OP, none of this is your issue. He shouldn't be dumping this stress on you. It's nothing to do with you. He sounds like he's bulldozed his way into your life and home (of his own free will by the way, it's nothing to do with his ex) leaving you no time to really think about anything. That's a red flag for abuse, OP.
"He lived with his parents while he got back on his feet" = he was homeless.
Also, he didn't get back on his feet, he moved in with you. That was his solution to his mess, dating "the boss's daughter" (or whatever relative you are) who's financially solvent and a homeowner, then wheedling his way into your home.
He's NEVER going to move out now unless you kick him out. Why would he? He has what he sees as a happy relationship with you and you now live together. Relationships don't go backwards, they generally only move forwards, he's not going to return to dating whilst living elsewhere. So if you kick him out you break up, basically.
Of course his DD needs a bedroom at her father's home, which happens to be the house you own. If you stay with him you're going to have to either give up your guest bedroom and guests can only use it when his DD isn't there or you give up the third bedroom that you're currently using as a walk-in-closet.
You need to decide what you want. If it's a "proper" relationship with a live-in partner or husband, but you don't like stepchildren (which is totally fine and doesn't make you evil or unfit to be a mother), then this man isn't it. If that's what your goal is, why date someone who isn't free for however many years it ends up being before his DD gets a place of her own? Some youngsters are in the family home until their 30s these days! You'd be limiting yourself unnecessarily by staying with him, if what you ultimately want is a child-free partner/husband.
On the one hand you're all "we" are doing xyz and you're living together, like you see you-and-him as a partnership. But on the other hand you're acting like it's your house and your life, he's just a boyfriend who is staying with you temporarily as a guest, until he buys his own place. That's not a realistic way to carry on because you're giving him mixed messages.
Decide which it is.
If you're a partnership, you see this as a "forever" relationship and he's moved in with you, then give his DD a bedroom because she's partly living there.
But if he's just a boyfriend who you've allowed to stay as a temporary guest and you're expecting him to get his own place soon, then communicate this. And tell him his DD can't stay overnight because this isn't his home and he doesn't live there himself, just is staying temporarily. So he can't allow his DD to partly live there either.
In the latter scenario, you could compromise by saying she can stay over only if it's convenient for you (he'll need to check with you each time), as a guest in the guest bedroom. So no leaving her personal items or clothes there, she comes with a suitcase and takes it all back home with her like any other guest would. Make it clear the third bedroom is your walk-in-closet and this is how you want it to stay.
Then stop getting involved in any of his custody stresses. If you're only dating as boyfriend/girlfriend and not a partnership, then he should be bringing his best self to your dates and you should be going out having fun, not bringing all his drama and stress about his "past" (it's actually his present) and offloading onto you on the daily, heavily using you as emotional support as if you were married. You should be having nothing whatsoever to do with his custody battle, his ex or his DD's parenting/childcare situation, if you're not a partnership.
You're taking on all his stress when you don't need to be. If he can't bring his best self to dates because of the level of stress he's under, then he needs to accept he's not actually in the right mindset for dating anyone at the moment, because he needs to focus on sorting his shit out.
You have boundaries by having them. You decide them and you have them, which means upholding them and accepting that other people may not like that. It means accepting that others feelings about it are none of your concern and not something for you to "fix" by backing down. It also means understanding that someone who repeatedly tries to trample your boundaries isn't a good person and shouldn't be part of your life.
It's ok if you want your house to feel like yours and not a shared-with-a-partner home. But you need to stop giving off mixed signals about it and stand up for what you want. If that leads to disagreements and the relationship ends, so be it, it means it wasn't the right relationship for you. If the disagreements lead to emotional abuse (sulking, silent treatment, put downs etc) or verbal abuse (shouting, name-calling etc) then accept you need to be the one to end what has become an abusive relationship. You're never going to be able to uphold your boundaries if you're not prepared to be single, because that mindset will lead to you backing down when you should stand firm, out of fear of the relationship ending.
By the way, his ex is not the monster he's portraying. She stops contact? How is that surprising when her DD doesn't want to go (at the time) and when DD's contact time with your BF is actually being spent at his parents place? She's stopped contact because she doesn't want her DD being used as his parents carer, which they sound like they need, and aren't in a position to provide care for his/her DD. That's her being a responsible parent.
I note that despite not paying rent or a mortgage, because he lives with you, he's also not paying for suitable childcare for his DD. His response to his parents being vetoed as suitable childcare has been to see his DD less. What an arsehole.
So she has family overseas? And took DD to visit for an extended period of time (probably because it's expensive and impractical to go more often for a shorter time period, or because she needed family support after the split)? Hardly surprising when he's a deadbeat dad who
a) still hasn't provided his DD with a bedroom of her own
and
b) was so shit a parent in the early years of DD existence that when they split, DD initially didn't want to spend time with him.
Why would she prioritise contact between your BF and his DD over spending time with her own family, in those circumstances?
I'm not seeing parental alienation, I'm seeing him pulling the wool over your eyes with a pack of lies and bullshit and woe-is-me type nonsense. Which the "dislikes his job" and "can't get his own place because it's his ex's fault" is part of it. He works for your family, the polite thing to do is to never mention anything negative about them and to accept that most adults of working age dislike their jobs. If he dislikes it that much the thing to do is look for another job, not complain to you about it (likely with a view to giving up work altogether and living off you).
If you want DC OP, pick another man, this one isn't a great partner or dad, so don't shackle yourself to him for life.