No, you don’t have to give him anything OP. the council can help him with that. He won’t pass a credit check anyway if he’s got no job, no way in hell. You have to demonstrate that you have an income of 2.5 times the rent in order to pass a credit check and he doesn’t, so if you don’t kick him out you’ll never get shot, because he’ll make sure he never gets a job.
It is also not your problem that he has a medical condition. If he wanted a safe place to live he should have made sure it was safe by not being a violent intimidating bully, and yes, slamming about and throwing things is violence.
Tell him to go and if he starts call the law. Turn him out and let the authorities deal with him. You don’t owe him anything.
OP I’ve been in a similar situation only it was I who couldn’t find anywhere to live and the council wouldn’t help. It’s miserable and dangerous to be trapped with an abuser, especially when they know it’s your intention to split. You are in danger from this man if you don’t get him out, whether you accept it or not, because when he feels his control slipping god knows what he might do, and you really don’t want either you or the DC to find out. In my case the slamming things and throwing things turned into being bodily thrown across the room and being threatened by a 16 inch kitchen knife, so beware. Please don’t put your DC in that situation and kick him out.
You want to be decent, I get that, but he isn’t decent to you and he’s not decent to your DC if he’s abusing their mother. Being the bigger person in this case is getting him out and protecting yourself and your DC, so do it.