It doesn't really add up to me.
You say you're keen to leave your high-poverty area... Well, you can! You don't need to hitch your wagon to this guy in order to do that. If you want your kids to grow up in a better area, then make that plan.
Or is it that you could only live in a better area with his income? If that's the case, then my advice: do not, do not , put yourself in this dependent position with a man who hasn't made you a serious offer and I mean marriage. Marriage means you are a team, responsible for each other, pulling in the same direction. You won't have to feel "a burden" as you will both have clearly signed up to live a shared life with shared assets.
But you don't mention his savings. With his high income and low living costs, hasn't he got any? You've done so well saving while raising two kids alone.
I don't think he's ready for marriage right now - what's he been working towards all this time? So you should keep things as they are until he is ready to step up. Do not let him be a man-baby who clicks his fingers and moves his girlfriend closer for his own convenience, while making no commitment of his own. Respect yourself - you've worked so hard - expect better from a man than this half-hearted, half-baked plan. I don't even understand it, all these moves. Has he heard of stamp duty, solicitor's fees, moving costs? The utter stress of making offers, getting a mortgage, things falling through? Moving house is a real upheaval, not a move on a chess board. If he can't face an hour's commute (which is nothing - I do that plus the school run on top!), is he seriously up for all that disruption? He hasn't thought it through, and those are your kids' lives he's not thinking through. And yours.
I think you've got a great deal right now. Boyfriend over half the time, not having him in your hair (or around your kids) the other half of the time. You don't need to be in any rush. You certainly don't need to be in any rush whatsoever to bear a child to a man who won't bother to marry you, when you already have two of your own. If he wants a child of his own, he needs to step up to the plate and be a proper husband and stepfather, not dither about making madcap schemes that leave you exposed.
You have more power than you think you have, and more strength. Don't give it away. Let him rise to the challenge. In the meantime, enjoy the present day.