Honestly not much more to discuss. It has been resolved at least to my satisfaction that there's really no clear way to get out of the situation I'd gotten myself in by the time the turn fell.
For the sake of giving the rest of the hand history, I did two even dumber things and got super lucky. Bet $45 on the turn and he called. River came an offsuit 5 and I bet $60 and he called. I rolled my hand and he mucked.
Has to be one in a hundred that this guy flats on both those streets with a worse hand, so not much to learn, particularly given this exact circumstance is like 0% to recur in my lifetime.