A doctrine doesn’t have to be designated as de fide in order for it to be sinful for denying it.

No problem here, BOD pertains to people who have reached the age of reason. Unbaptized infants can’t be saved without the sacrament of baptism.

Please read it again.  It's not just about infants.  He's talking about each an everyone who desires Baptism forfeiting the Kingdom.  Infants do not desire Baptism.  Pope says either infants "OR" those desiring it.

And the way you express this speaks to the grave error floating around.  When you say unbaptized infants can't be saved without the Sacrament of Baptism, the implied corollary is that adults can be saved "without the Sacrament of Baptism".  This is in fact a heretical proposition.  NO ONE CAN BE SAVED "WITHOUT" THE SACRAMENT OF BAPTISM.  In all cases, the Sacrament of Baptism is the instrumental cause of justification.  At best you can say the Sacrament can be received in voto, but not that anyone is saved WITHOUT the Sacrament.

In fact, this is probably the top argument against the BoD interpretation of Trent.  Same thing applies.  If you claim that this is an either ... or proposition, that justification cannot happen without the Sacrament or else the Votum for it, there's an implied heretical corollary that one CAN be saved "WITHOUT" the Sacrament.

I could go into other problems with the BoD interpretation of Trent, but I'll leave it here because this point dovetails with the implied heresy in your statement.

So the teaching in Trent can in theory have one of two senses:

1) I cannot write a letter without a pen or a pencil.  This means I can write a letter with one OR the other, and it also means that I CAN write a letter without a pen (so long as I have a pencil) or vice versa.

2) Wedding cannot take place without the bride or the groom.  In this sense, both are required.  If one OR the other is missing, the wedding cannot take place.

In sense #1, there's the implied corollary that it CAN happen WITHOUT one or the other, and therefore that justification CAN happen WITHOUT the Sacrament of Baptism.  Problem there is that Trent explicitly condemned that proposition as heretical.

Finally, right after the passage, Trent adds, [paraphrase] "Our Lord taught that one must be born again of water AND the Holy Spirit."

Water corresponds to the laver, and the Holy Spirit to the votum, which as Catholic Encyclopedia even states, entails much more than a simple desire for Baptism, but rather all the dispositions required to be baptized unto justification.  And Trent had just spent several paragraphs explaining that it is the Holy Ghost who inspires and motives these dispositions.

So the have Trent adducing this as proof text for the either ... or reading would be tantamount to saying ...

"We cannot play baseball without a bat or a ball, since Bob says that we need a bat and a ball to play baseball." ... saying that this sentence mean that we can play baseball with EITHER a bat OR a ball.

No matter which angle you take, the BoDer reading of Trent fails.

Finally, the problem here is that nowhere does Trent positively teach BoD, even IF you accept the either ... or interpretation.  At best, Trent is saying that it's erroneous (heretical) to hold that the Sacrament is not necessary AT LEAST in desire.  This means that Trent would be leaving open the BoDer interpretation as non-heretical, but not positively teaching it as true.  When Trent intended to teach something in a positive way, there's a Canon at the end reinforcing the teaching.

It would be similar to me, if I were unsure about BoD, saying, that Baptism is necessary at least in desire.  One poster here who's a BoDer, actually reads Trent this way, that it's just saying that you must hold that Baptism is necessary at least in desire, but is not necessarily teaching positively that desire suffices.