L I am both a lawyer (although not family law) and had a husband rather like yours (although he didn't as far as I know lie to anyone about me like that). He did not want a divorce.
This is what I did - first went to see a solicitor on my own without him knowing and paid for an hour of advice. The one thing I wanted to be sure about was whether I would lose the children if we split up and she said not (the older teenagers would have and did choose to live with me). We both worked full time.
Then he was advised to stay in the home (as I would advise you by the way however horrible the husband is) until the divorce was final. It was an awful 7 months for me (and presumably him) with a lot of nastiness from my husband but it was worth it.
You need to find out of your mortgage company would transfer the mortgage into your name i.e. if your salary is enough to support that mortgage. Mine was. So part of our divorce financial order (which the court approved) was transfer of house to me and a cash sum to my husband - I did a big remortgage. If the house is in negative equity and the family has no other money as your position might be then it is more likely the house and mortgage would have to stay in joint names but your husband moves out on the divorce and possibly but not certainly as it is not his child, he pays child maintenance (probably not maintenance for you as you and he still work and you would have to pay all the mortgage but this is just everyone's comments on the thread, a solicitor with all the information about who earns what etc would be better able to advise). The child maintenance is likely to be 15% if his net pay with deductions for the time the child stays at him at his new place. The child is probably counted as a "child of the family" even though it is just his step child (and indeed you might be able instead to get maintenance from the child's father but that's a different story.
If your husband earns a lot more than you do he may have to pay you spousal maintenance too.
Anyone can check their house value - go on line and look at Land Registry sold prices for similar houses recently in the street and also look at what your mortgage value is. Also check what yours and his (if any) pensions are as that is another family asset some people have.
My guess is the most likely outcome is the house remains in joint names and mortgage, he moves out and the child is mostly staying with you. As you say you might get tax credits. Then the financial court order which by the way you and he can agree instead of court hearings if you both can (my husband and I did that although both of us had solicitors), is likely to say you remove him from the mortgage as soon as your earnings are high enough and that the house is sold as soon as the child is 18 or you remarry or cohabit. I know it's not his child but the Children Act I think talks about a child of the family. I knew one man who was paying £60k a year school fees for his step children to attend Millfield school I think it was, a boarding school and after the divorce he was forced to continue to do that even though they were not his children but because he had been supporting them whilst they were living at home with their mother, his second wife.
We are all pretty good on the financials and legals but for me actualyl the hardest thing with a man like this is the psychological and emotional issues. I had to tell myself as the lawyers told me ultimately he could be forced out of the house as I could afford to buy him out but he could not do the same for me so he WOULD be going. Had he not moved out the day after he was removed from the mortgage and money transfer to him we could have had a court order to remove him. The sequence was sending draft divorce petition to his lawyers to check/approve, then decree nisi, then agreeing financial / children arrangements, court sealed that "consent order", then decree absolute, then mortgage transfer into my sole name/remortgage and cash transfer to him and then he moved out. That took us 7 months - no court hearings as we reluctantly and with a lot of rowing agreed the figures in the end.
I think it remains hard to get a court order excluded a man or woman from the marital home when there is no violence but I am not an expert. There was a case just this week on coercive control by a man (with a very light sentence by the way - ugh) not in a divorce context - that is a new area of law - and it might be more constructive to negotiate divorce and separation terms with him or if he won't agree them pushing it ahead quickly to court hearings even if you represent yourself. However if a solicitor advises you might be able to get him excluded from the home that might be a second possible intermediate solution - even there though ultimately you would want everything fully settled.