In a 2 1/2 bedroom home where the 1/2 bedroom is to be an office, OPs 2 DC will already be sharing a bedroom. There will literally be no space (except a sofa?) for stepson.
I think you're getting a hard time OP. The boy has basically moved full time to his mother's home and shows almost no interest in his father. I'm not surprised DH didn't foresee him living with you. I don't agree with all the just-in-case stuff. If stepson's mother died DH could worry about how to house him then, he could be on the sofa temporarily until something more suitable was sorted out. In reality his mother isn't terminally ill so unlikely to die in the next two years.
Stepson's mother's benefits will reduce when he's 18. Is she planning on kicking him out, or asking him to pay rent, then and that's what has got him thinking he'd rather live with you (where he possibly presumes he'll continue to pay nothing)?
As others have said, if DH can't afford to house him as an adult either then a conversation needs to be had with stepson about his future plans. If a larger house is needed maybe he's going to have to pay towards the expenses of that. Once he realises he can get a house/flat share for not much more per month, and that it'll come with a whole heap more freedom then living with a parent does, he might prefer that option. So you'd have bought a bigger house for nothing and be left with all the costs yourselves.
The holiday without him I'd have told him it wasn't a your-nuclear-family type holiday like usual. It was a gift by your parents for their children and grandchildren. So you and your sister, plus your DC and their cousins. The spouses coming along was just because they can share a room with their partners, so no extra expense. I think if you'd worded it like that he'd have seen that not inviting him was no different to your parents not inviting his half sister who lives with his mum, neither of them are your parents grandchildren.
Stepson seems to be picking and choosing which bits of blended family life he wants. That's unrealistic and confusing for everyone else. I think it could be a useful lesson for him (if you sit down with him and talk about it) that if he wants to be part of a family he needs to, well, be part of it. Not pick and choose the most fun parts and opt out of anything he sees as drudgery.
At the moment that's the message both his parents have given him. He's at risk of growing up to become a partner who does the same thing he's doing now and opts out of family life when it's too boring, leaving majority of the grunt work to his partner.
If he wants "in" to your family, he needs to get on board with being part of birthday celebrations always, or at least be concrete on his availability and not leaving it to last minute to decide. That is him treating the birthday event as an option if he's got nothing better to do. It's rude to the birthday person.
He should be doing house chores, earning his own spending money with a part time job, eating meals at the table with the rest of the family, going on family days out with you all. And coming over and staying regularly, if he wants to maintain a bedroom in your home. Maybe not 50/50 if that doesn't suit him any more, but EOW or something.
He can't really go about insulting his younger half siblings, ignoring special events unless there's a freebie in it for him, barely visiting, refusing to spend any time with his father out of embarrassment or whatever and then still want to be considered "part of the family".
You haven't pushed him out, he's removed himself for his own teenage reasons. He's old enough now though to start learning you reap what you sow. I wouldn't hold anything against him up to now, he's young, he's learning about life and we all make mistakes. But I'd be having a conversation with him about whether he wants "in" (and will put the effort in with family relationships) or wants "out" (to go into the world as an adult, forge his own life and come back to visit occasionally), because he can't keep on cherry picking the bits he wants like he is currently doing.