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These two passages from Trent cited by the Dimond Brothers completely destroy the notion that there can be [initial] justification without remission of all punishment due to sin:

Simple syllogism.  Initial Justification (vs. justification of the fallen in Confession) requires being born again.  Being born again means that there is nothing in someone that God hates so much so that "nothing may delay them from entry into heaven" (echoing Innocent III here)

In His infinite mercy God has willed that the effects, necessary for one to be saved, of those helps to salvation which are directed toward man's final end, not by intrinsic necessity, but only by divine institution, can also be obtained in certain circuмstances when those helps are used only in desire and longing. This we see clearly stated in the Sacred Council of Trent, both in reference to the sacrament of regeneration and in reference to the sacrament of penance (<Denzinger>, nn. 797, 807).

https://www.ewtn.com/catholicism/library/letter-to-the-archbishop-of-boston-2076

For, it was not to private judgments that Our Savior gave for explanation those things that are contained in the deposit of faith, but to the teaching authority of the Church.

It is evident, both from Bellarmine and Billuart, that no one can be saved unless he belongs to the visible communion of the Church, either actually or virtually, and also that the salvation of catechumens can be asserted only because they do so belong ; that is, because they are in the vestibule, for the purpose of entering,  have already entered in their will and proximate disposition. St. Thomas teaches with regard to these, in case they have faith working by love, that all they lack is the reception of the visible sacrament in re ; but if they are prevented by death from receiving it in re before the Church is ready to administer it, that God supplies the defect, accepts the will for the deed, and reputes them to be baptized. If the defect is supplied, and God reputes them to be baptized, they are so in effect, have in effect received the visible sacrament, are truly members of the external communion of the Church, and therefore are saved in it, not out of it. *(footnote: * Summa 3, Q. G8, a. 2. corp. ad 2. et ad 3.)



Bellarmine, Billuart, Perrone, &c, in speaking of persons as belonging to the soul and not to the body, mean, it is evident, not persons who in no sense belong to the body, but simply those who, though they in effect belong to it, do not belong to it in the full and strict sense of the word, because they have not received the visible sacrament in re. All they teach is simply that persons may be saved who have not received the visible sacrament in re ; but they by no means teach that persons can be saved without having received the visible sacrament at all. There is no difference between their view and ours, for we have never contended for any thing more than this ; only we think, that, in these times especially, when the tendency is to depreciate the external, it is more proper to speak of them as belonging in effect to the body, as they certainly do, than it is to speak of them simply as belonging to the soul; for the fact the most important to be insisted on is, not that it is possible to be saved without receiving the visible sacrament in re, but that it is impossible to be saved without receiving the visible sacrament at least in voto et proximo, disposition.

http://orestesbrownson.org/210.html

False.  You laid out this fanciful narrative about temporal punishment due to sin not being remitted by BoD.  Since you made these assertions, you prove them.  I can't and don't have to prove a negative.

Show me a single proof for your made-up narrative about temporal punishment due to sin not being forgiven by BoD.

You won't find any because there isn't any.  This is completely made up out of thin air.

Now, we do know that St. Alphonsus held this opinion, so my criticism of your post is at the same time a criticism of St. Alphonsus.  There is no proof that BoD does not remit temporal punishment due to sin.

I'll find the video from the Dimonds on Trent.  As Our Lord taught, one cannot enter the Kingdom of Heaven unless one has been BORN AGAIN.  Rebirth means a complete regeneration (as Trent defines it also), including remission of all temporal punishment due to sin.  Trent makes that clear.  So if there's such a thing as BoD, it must be a rebirth or regenerations, and thus it must remit all temporal punishment due to sin.  This is yet another error made by St. Alphonsus on this matter.  I believe that St. Alphonsus was a bit too enamored of some Jesuits in his day, such as De Lugo, and even grants the latter's opinion regarding the possibility of salvation for infidels as "probable" (their word for "possible") ... though not holding it himself ... just because he had a high opinion of De Lugo.  But De Lugo's opinion was horrible and rejected 1500 years of teaching that explicit knowledge of Christ and the Holy Trinity are necessary for salvation.

Now, this was before the infallibiilty of the OUM had been defined by Vatican I, but if a teaching that was unanimously held and taught by the Fathers, and by all Catholics, for 1500 years is not infallibly taught by the OUM, then there's no such thing as the infallibility of the OUM.  Nobody doubted this teaching for 1500 years until a Franciscan and a few Jesuits came along and complete made up "Rewarder God" theory out of thin air ... so they could get the newly-discovered Native Americans saved somehow.

If it's not OK for us to reject BoD on the grounds that nearly all theologians have held it for the last 400 years, then why is it OK for these guys to come along and reject 1500 years of teaching to the contrary of their opinion?

Also, the Holy Office upheld the teaching that knowledge of the Holy Trinity and the Incarnation are necessary by necessity of means for salvation.