Thanks everyone for all the suggestions and messages of support.
Bit of an update – my baby (now 12 weeks!) seems to be getting the hang of being put down in her carrycot at night which is wonderful.
Tried a load of the suggestions up thread, but what seems to have worked for her (if anyone is reading this looking for suggestions!) is waiting until she is in deep sleep, checking with the arm drop test, placing her blanket (which smells of me as I snuggle with her and it during the day) around her back and letting it warm up so she's not being put down on the carrycot but onto the blanket, and only when she's deeply asleep with the blanket on her back am I putting her down. I'm sleeping on the sofa with the carrycot right next to me, so she knows I'm right there, and am able to replace her dummy the moment she drops it and shush/pat her if she stirs at all and I think that makes her feel more secure. It all seemed to click suddenly into place, and she accepted being put down more and more from a week or so ago, and the past three nights has slept in the carrycot from 11/12 at night until 8/9 in the morning which is amazing (barring feeding quite a lot, I think she's still got quite a little tummy but she's fairly small for her age so I guess that makes sense!).
It's also possible that she might have been experiencing silent reflux caused by dairy as I cut dairy out a few weeks ago and I know it takes ages to leave their system, so it could be that. Don't know if she was reacting to dairy or not, to be honest, but dairy is definitely not worth the risk so will just avoid it until she's weaned.
I'm well aware she's not been doing it for long so it might not last, but I got the longest stretches ever of sleep in the past few days, as did my DH who got solid 10h stretches of undisturbed sleep in bed both days this weekend, so we're both feeling quite refreshed.
My mental health referral has come through really quickly and I've got my first appointment Friday coming, which is much sooner than I expected. My mental health has been loads better since starting to get a little bit more sleep – it was EXTREMELY poor a couple of weeks ago to the point where I was worried I'd end up suicidal, but I think that it was entirely caused by severe exhaustion because I'm feeling basically 100% fine since being a bit more rested! It was crazy to go from sobbing/hyperventilating for hours through the night to sleeping and feeling completely calm in the space of a couple of days but I really do think it just shows how important sleep is for mental health.
Particular thanks for all the advice on safe co-sleeping, as once she's in a really good routine of being happy lying on her back, I'm going to try the next-to-me crib again, and will be happy to co-sleep with her in the bed if she won't go down in the next to me. Would really like to sleep in a bed again as I feel quite stiff after sleeping exclusively on our (very small!) sofa for so many days running, but so long as I'm getting sleep, I'm not complaining.
I also spoke to my parents about how hard I'm finding things and we've agreed to all quarantine completely for a fortnight after the baby's jabs this week coming so we know we're all covid free and then I'm going to stay with them for a couple of days for some extra support. Really looking forward to that. And just before I start quarantining, I've got a socially distant walk with a friend planned, which is going to be the first time I've socialised with someone except my husband since the 5th December, so I can't wait. Think I've been being a bit too cautious re: covid to the detriment of my mental health.
Sorry for the essay and for not replying to every poster individually; I really truly am grateful to each and every one of you who replied at a time where I was in such a dark place. I listened to you all and have looked at/will be following up on a number of the suggestions!