So, doesn't that mean that he and St Robert Bellarmine (Doctor) are at odds? How is this squared?
Well, logically, one of them was wrong. As is clear from an objective look at the evidence, some were for the notion, some against. St. Augustine was both for AND against, having floated the idea when he was younger but then rejecting it later.
That's why I say that the status of BoD is that it's a controverted matter among Catholics. There's no indication whatsoever that it was revealed, and so it may never be resolved.
On another thread, however, I posted that, if I were pope, I would fall short of condemning BoD as heretical but would forbid all discussion of it, since it does no good but lots of harm. You see, if God does save by BoD, then not believing in it won't change that. In fact, ironically, if someone ardently believes in BoD, that could actually UNDERMINE their desire to be baptized, making them complacent that they can be saved without Baptism ... as Fr. Feeney famously pointed out. On the other hand, BoD very quickly leads to religious indifferentism and denial of EENS, and it is the root cause of all the Vatican II evils.
Here's my perspective. If I die and find out that God saved some people by BoD, then glory to God. But until then, the only thing I know for sure that He revealed to us is that the Sacrament
in re is necessary for salvation, and BoD has a tendency to completely undermine that dogma.