Per the title of this thread...Trent does not "teach" BOD. Very true.
It mentions "desire of baptism" but it doesn't teach or explain what that is, who qualifies, how it must be believed, etc.
Every other doctrine defined by an ecuмenical council CLEARLY and UNMISTAKENLY explains those doctrines in detail, with much explanation and theological thoughts. BOD isn't at all in the same category as defined doctrines for this simple reason alone.
A doctrine defined by a council should stand on its own. One should be able to read the council docuмents ALONE and understand what the teaching is. The simple fact is that no one can do this. Everyone who tries to explain BOD cannot use Trent, except to say it mentions "desire". To explain BOD, one has to give countless #s of quotes from saints and stories from history. St Thomas and St Augustine are more explanatory than Trent BY FAR.
When one cannot use a council to explain a supposed doctrine, then that's a sign it was never defined.