Don't see why you wouldn't be allowed that at all, if it's normal for the council to offer those free rent weeks, and you're allowed to overlap for 28 days. That seems a really positive compromise - you have a nice new place to move to, and you have 6 weeks grace in the refuge to benefit from the support and sense of sanctuary there, too. Don't tell the council it will depend on his behaviour, just say yes you want it and sign. If he behaves badly enough that you can't move in, then they will just have to let it to someone else - you're allowed that 28 days overlap so you can explain why you need to break the contract and as you say, someone else will leap at the chance of it. I don't see why that isn't doable.
Also bear in mind that unless his solicitor is totally stupid, she will have to tell him that he is risking his volume of contact with dd if he steps out of line again with you. Up until now he's been his mummy's spoiled little prince who is allowed to behave any way he wants without consequence, and so he's bullied, threatened, stolen from and abused you without seeing any reason why he shouldn't. That isn't how a court will see it, and any good solicitor will warn him that he has fucked up badly already and any more abuse from him will make that worse. You're starting this contact hearing offering contact in a safe setting with a background of needing state funded legal aid and a place in a women's refuge because of him, and any decent lawyer will impress upon him that this is not a position of strength for him, and he needs to be Mr Perfect, going forwards. Hassling you will not help his case. And you were going to talk to the solicitor about a non-molestation order, weren't you - how was that going?
You have the offer of a lovely new home, that (being council) nobody can take away from you, and which you can make safe with advice from the shelter and the police. He doesn't know where it is and has no way of finding out, especially if you get a new car, as mentioned. You have a solicitor now to help protect you, and you should be able to get social services support if you're in a shelter as well. These are all really good things, Far, so you can protect your daughter and offer her a lovely new future moving onwards. He can huff and puff but he doesn't even know where your new home will be - he can't blow it down. You're safe and sound where you are and you can make the new place safe and sound, too, with advice on how to do that. It could be a real home for you and your dd, and you can from the sounds of it take that time to be supported in the refuge first, too. You're doing great.