Sounds like he'd rather ignore the problem than address it, or sleep in damp / smelly sheets than admit he's wet the bed.
The way your wroting about it is purely in terms of external actions and reactions, which seem to miss the idea that humiliation and embarassment can be instrinsic... He's not a program where inputs = outputs. He's a boy with his own thoughts, feelings and reactions which come from inside of him. So don't ignore him feeling feelings that you haven't given him
eg just because (you think) you haven't done anything to make him embarrassed, he isn't embarassed.
He's too old to see the situation in this way. His influences will be from inside himself, as well as the vast and persuasive society and culture he lives in, from class mates, teachers to tv and media, he will be receiving and internalizing a massive amount of stuff about what it means to still wet the bed.
Nb also by understanding the situation as being about you messages in = behaviour out, you are making yourself entirely responsible - way to feel massive guilt!!!
And just because he isn't able/ willing to articulate it verbally, doesn't mean to say he's not feeling very complex feelings about it, that are coming out as stubbornness & refusal to engage.
I would say shutting down about it completely is a protective mechanism rather than the reality. Certainly not a sign he doesn't care, in fact it's a classic way of dealing with something you feel you can't fix isn't it? To pretend you don't care, or are even doing it on purpose!
Maybe he can't deal with it or feels he can't change, so pretends to himself it's not happening?
So he needs help to cope with it happening, vs being forced to break through his way of protecting himself (which is a rubbish way, not saying it isn't!)
Sorry typing whilst traveling, hope some of it makes sense