You have other options OP.
If you have access to money, stay put, scrimp on everything and save up for a deposit on a private rental, or even better 6mths rent upfront so you don't need a guarantor then. Move into private rental. Use housing benefit to save up for the next 6mths rent. If it goes tits up at any point, you apply as homeless again to the council. Phone the current council back and tell them you want to join the list anyway even if you're "priority z" and would realistically get housed ten years after you've died of old age. Once you're on the list your application can be upgraded or downgraded as necessary due to your circumstances. How long you've been on the list is taken into account when deciding who to allocate a property to.
If you don't have access to money then there's absolutely no doubt you're being financially controlled, contact Women's Aid. In fact do that regardless, let them decide if your situation is abusive and if they'll help. Tell them you're unsure if abusive but others can see it and have told you so. I know what people are saying with temporary accommodation being grim, tell everyone about your mental health too (including GP if you haven't already) it can help you get appropriate accommodation. But sometimes physically grim is easier than mentally grim. You can clean, you can try to avoid neighbors, ignore their noise etc. It maybe not so easy to avoid being ground down by emotional abuse if you stay with your partner. I'm worried about escalation of abuse too. You hate him, you presumably don't want to sleep with him, if you refuse will he pressure you or get violent? It's one thing saying you can stay but does he mean as a girlfriend or just as someone needing a home? Not wanting you to leave, not wanting be without you, wanting you to stay, wanting to work things out: can all be code for wanting to continue to abuse you (if he is being abusive), or just wanting to keep you in a shitty disrespectful relationship because it suits him, or not wanting to pay child support. It doesn't necessarily mean he loves you or wants the best for you or baby or that you're compatible.
If Women's Aid can't help, you can go to your mum's and claim benefits. Tell her it's for one night. Then refuse to leave. Stay out of her way as much as possible. Sit in a room she's not in, keep baby quiet, go for lots of walks, do house chores, try not to be a nuisance at all. One of two things happens. You either save up enough for deposit/6mth rental in private flat and go. Or mum loses her temper. At that point you apologies and say you had to do it because you can't make yourself intentionally homeless or you'd be on the streets and your baby in care. Ask mum to write a letter saying you're kicked out with immediate effect, and apply for homeless with the council.
Get a job, any job anywhere in the UK. You now have a reason to apply for the council list in that area. You don't have to be homeless to get on the list. Go there and update the council that you're homeless. Go to the job even if you have to take the baby with you and get fired. Show people you're trying. You might not get fired anyway, colleagues could know of nurseries etc and help you. This could also work if you've any friends in another area who could take you in temporary. You then live in that area so have reason to join the council house list.
You could throw yourself on the mercy of his parents, your child's grandparents, who might take you in while you claim benefits and save up for somewhere.