Saint Augustine’s view on the divine consignment of unbaptized infants to the positive sufferings of the damned in hell, contra S. Aquinas, and in addition to the lack of the Beatific Vision, is found in the following resources:

De pecc. mer. 1.16.21 (CSEL 60, 20f.) ; Sermo 294.3, Patrologia cursus completa, series latina (PL), J.P. MIGNE (ed.), 38, 1337; Contra Iulianum 5.11.44 (PL 44, 809).

For your convenience:

https://archive.org/details/sanctiaureliiau05augugoog/page/n659/mode/1up

Chapter 16 [XIII.]— How Death is by One and Life by One.

And from this we gather that we have derived from Adam, in whom we all have sinned, not all our actual sins, but only original sin; whereas from Christ, in whom we are all justified, we obtain the remission not merely of that original sin, but of the rest of our sins also, which we have added. Hence it runs: "Not as by the one that sinned, so also is the free gift." For the judgment, certainly, from one sin, if it is not remitted — and that the original sin— is capable of drawing us into condemnation; while grace conducts us to justification from the remission of many sins — that is to say, not simply from the original sin, but from all others also whatsoever.

3. This is the first error that needs to be turned away from people's ears, and uprooted from their minds. This is something new in the Church, previously unheard of, that there is eternal life apart from the kingdom of heaven, eternal salvation apart from the kingdom of God. First consider, brother, if you shouldn't perhaps agree with us on this point, that whoever is not consigned to the kingdom of God is undoubtedly consigned to damnation. The Lord is going to come, and pass judgment on the living and the dead, as the gospel says, and to make two groups, on the right hand and on the left. To those on the left he is going to say, Go into the eternal fire, which has been prepared for the devil and his angels (Mt 25:41); to those on the right he is going to say, Come, you blessed of my Father, receive the kingdom which has been prepared for you from the origin of the world (Mt 25:34). On this side he mentions the kingdom, on that damnation with the devil. There is no middle place left, where you can put babies.

Judgment will be passed on the living and the dead; some will be on the right, others on the left; I don't know any other destiny. You there, bringing in a middle place, get out of the middle, don't make the person seeking the right hand trip over you. And I'm advising you for your own sake; get out of the middle, but don't go to the left. So if there will be a right hand and a left, and we know of no middle place in the gospel; here on the right hand is the kingdom of heaven: Receive, he says, the kingdom. Whoever isn't there, is on the left. What will be happening on the left? Go into the eternal fire. On the right to the kingdom, eternal of course; on the left to the eternal fire. Whoever is not on the right, is without a doubt on the left; so whoever is not in the kingdom is without a doubt in the eternal fire.

Can those who are not baptized really have eternal life? They won't be on the right, that is they won't be in the kingdom. Do you count everlasting fire as eternal life? And about eternal life itself, listen to a more explicit statement that the kingdom is nothing else but eternal life. First he mentioned the kingdom, but on the right; eternal fire on the left. In the final sentence, though, to teach us what the kingdom is and what eternal fire is, Then these, he says, will go off into eternal burning, the just, however, into eternal life (Mt 25:46).

There you are, he has explained to you what the kingdom is, and what eternal fire is; so that when you confess that a baby won't be in the kingdom, you are admitting it will be in the eternal fire. The kingdom of heaven, you see, is eternal life.
 

Book 5

[ . . . ]

Chapter 11

(44) You quote from the Gospel: ‘It were better for that man if he had not been born.”* But was his birth not due more to the work of God than his parents? Why did not God, foreknowing the evil that lay before him and which parents cannot know, give the better portion to His own image? Those who understand rightly know that nothing is attributed to God except what is proper to the goodness of the Creator. In like manner, without any difficult investigation, we must attribute to parents their wish to have children, although they know nothing of their future. But I do not say that children who die without the baptism of Christ will undergo such grievous punishment that it were better for them never to have been born, since our Lord did not say these words of any sinner you please, but only of the most base and ungodly. If we consider what He said about the Sodomites, which certainly He did not mean of them only—that it will be more tolerable for one than for another in the day of judgment,? who can doubt that nonbaptized infants, having only original sin and no burden of personal sins, will suffer the lightest condemnation of all? I cannot define the amount and kind of their punishment, but I dare not say it were better for them never to have existed than to exist there. But you, also, who contend they are, as it were, free of any condemnation, do not wish to think about the condemnation by which you punish them by estranging from the life of God and from the kingdom of God so many images of God, and by separating them from the pious parents you so eloquently urge to procreate them. They suffer these separations unjustly, if they have no sin at all; or if justly, then they have original sin.

Thanks. I think this is the translation of that:


Chapter 21 [XVI.]— Unbaptized Infants Damned, But Most Lightly; The Penalty of Adam's Sin, the Grace of His Body Lost.

DecemRationis, it seems like one long game of the telephone game, but I'm struggling to find it's source.

If you search Augustine, infants, and the fires of hell, many authors are merely repeating the claim without even bothering to list a source, even works from so-called scholars, which are nothing of the sort.

I'm now wondering if the claim that St. Fulgenitus also held that infants suffer the fires of hell is likewise a spurious claim.

It is clear that Bellarmine found the situation embarrassing, being unwilling, as he was, to admit that St. Thomas and the Schoolmen generally were in conflict with what St. Augustine and other Fathers considered to be de fide, and what the Council of Florence seemed to have taught definitively.

DZ693  Moreover, the souls of those who depart in actual mortal sin or in original sin only, descend immediately into hell but to undergo
punishments of different kinds.

If the knowledge of what has been hitherto explained be, as it is, of highest importance to the faithful, it is no less important to them to learn that the law of Baptism, as established by our Lord, extends to all, so that unless they are regenerated to God through the grace of Baptism, be their parents Christians or infidels, they are born to eternal misery and destruction.

http://www.catholicapologetics.info/thechurch/catechism/Holy7Sacraments-Baptism.shtml

There's nothing spurious about it.  I read the Latin posted here earlier, and St. Augustine clearly says that infants go to the place of ignis aeternus, eternal fire.  Now, maybe he felt they were in the place of eternal fire but didn't actually somehow directly burn in the fire itself, perhaps off in a corner somewhere, but it's likely that his vision entailed them getting singed in a mild way by the inferno.

Yes, we can all speculate - e.g, your "it's likely . . . " For example, St. Robert Bellarmine was of the opinion that St. Augustine did not hold that the infants were burned.

That is why I want to consider what he actually said and thought, as he himself expressed it.

Neither of these theologians [Bellarmine or a guy named Petavius], however, succeeded in winning a large following or in turning the current of Catholic opinion from the channel into which St. Thomas had directed it. Besides Natalis Alexander (De peccat. et virtut, I, i, 12), and Estius (In Sent., II, xxxv, 7), Bellarmine's chief supporter was Bossuet, who vainly tried to induce Innocent XII to condemn certain propositions which he extracted from a posthumous work of Cardinal Sfrondati and in which the lenient scholastic view is affirmed. Only professed Augustinians like Noris and Berti, or out-and-out Jansenists like the Bishop of Pistoia, whose famous diocesan synod furnished eighty-five propositions for condemnation by Pius VI (1794), supported the harsh teaching of Petavius. The twenty-sixth of these propositions repudiated "as a Pelagian fable the existence of the place (usually called the children's limbo) in which the souls of those dying in original sin are punished by the pain of loss without any pain of fire"; and this, taken to mean that by denying the pain of fire one thereby necessarily postulates a middle place or state, involving neither guilt nor penalty, between the Kingdom of God and eternal damnation, is condemned by the pope as being "false and rash and as slander of the Catholic schools" (Denz. 526).

This condemnation was practically the death-knell of extreme Augustinianism, while the mitigate Augustinianism of Bellarmine and Bossuet had already been rejected by the bulk of Catholic theologians. Suarez, for example, ignoring Bellarmine's protest, continued to teach what Catharinus had taught — that unbaptized children will not only enjoy perfect natural happiness, but that they will rise with immortal bodies at the last day and have the renovated earth for their happy abode (De vit. et penat., ix, sect. vi, n. 4); and, without insisting on such details, the great majority of Catholic theologians have continued to maintain the general doctrine that the children's limbo is a state of perfect natural happiness, just the same as it would have been if God had not established the present supernatural order. It is true, on the other hand, that some Catholic theologians have stood out for some kind of compromise with Augustinianism, on the ground that nature itself was wounded and weakened, or, at least that certain natural rights (including the right to perfect felicity) were lost in consequence of the Fall. But these have granted for the most part that the children's limbo implies exemption, not only from the pain of sense, but from any positive spiritual anguish for the loss of the beatific vision; and not a few have been willing to admit a certain degree of natural happiness in limbo. What has been chiefly in dispute is whether this happiness is as perfect and complete as it would have been in the hypothetical state of pure nature, and this is what the majority of Catholic theologians have affirmed.

Then it continues:
So, for the past several hundred years now, there are practically no theologians who have held that unbaptized infants go to Hell and suffer anything at all.  Why is it OK for individuals to reject this, including the same ones who keep arguing the "450 years" in support of BoD?

I hold the same opinion that was described that Suarez held above.

But, despite several hundred years of near-unanimous theological opinion in favor of Limbus Infantium, I do not claim that someone is condemned as a heretic for rejecting that Limbo, since it's not been actively taught by the Magisterium nor has the opinion of St. Augustine been condemned.  I'm consistent there.