OP wants to know how to fix the relationship with DS1.
The fact is, her prioritising DS2 over DS1 is what’s damaging the relationship. Yes, she is a single mother doing her best, and yes, DS2 can’t be left to harm himself, but also, yes, DS1 is less of a priority to her, DS1 only gets attention when DS2 allows it, and it seems like the house revolves around keeping DS2 calm while DS1 just has to fit in.
No, he doesn’t want to go to a group that revolves around ‘people with embarrassing siblings who take up all their parents’ attention and ruin holidays’ (as he probably sees it) and have yet another thing that’s centred around his brother. And he’s pushing away his mum when she reaches out because it seems insincere and forced, he feels like second best, and he resents it.
His feelings are entirely valid and accurate, just as OP’s feelings of being at her wit’s end are valid.
Is there an easy solution? No, not unless their dad, or other extended family members step up, and DS2 is assisted in adjusting to having others around when he’s dysregulated instead of OP (if OP died, he’d have to get used to that, so it’s not impossible).
I would suggest breezily ignoring him mouthing off silently or smirking, as annoying as it is – he’s looking for a reaction anyway. Sure, he’s being stroppy, but given the situation, I think it’s better to pick your battles and let that slide at the moment.
I do think an honest chat might help. Lay it out, and be a little vulnerable. Be genuine – tell him that it’s a shitty situation that is no one’s fault, it’s really tough on him right now, you know, and it’s tough on you too, but you just have to slog through it and hope things balance out soon. Etc, etc. Perhaps ask him to have a think about what he wants, what you can do for him, or things you can do to make the household dynamic better for everyone.
Right now he’s powerless in this situation, and perhaps having a say and being properly listened to might help. although it might take him a while to figure out what he wants to say.
It does sound like there’s a lot of built up stress and resentment that’s boiled over for him, though, and it might take him some time to get past this and stop reacting by pushing you away, even if you do everything right (especially as part of this will just be teen angst).