Thank you @SL2024. It's nice to know my efforts aren't entirely wasted. We are saving for our own home and don't have ownership over bills where we are, hence our personal circumstances being more complicated than others and older folks on here quick to judge. I'm sure we're not the only ones reliant on kind family as a stopgap.
In the end, in the mess of everything, it turns out we didn't pay our fee for my green card when we thought we did. USCIS sent us a payment notice, and it was the proof of address we absolutely needed to get everything rolling. We have a joint bank account now, I now have a State ID, and I've passed through my US driving theory yesterday all off the back of it. Mistakes into miracles.
I would suggest anyone in a similar position getting set up this way not to pay the green card fee until a notice is sent. It doesn't even delay the green card, which I also now have.
Regarding HMRC I reported not having my P45 and explained why. They will ask at the end of the form if you go through it online regardless of the warnings. It took some time and an error on their part making me resubmit the form again, but I got a message on my online account a few days ago saying the tax has been calculated and they owe me half a grand. They should be sending a cheque in the post. (Here or within the UK remains to be seen.)
It all got sorted out in the end. But they didn't make it easy. What happened with my employer refusing to give me my paperwork, who's to say. I can't say I care anymore.