Living together is untenable and creates a very bad environment for your children. You both need to acknowledge that and do everything you can to end this situation. You both have a responsibility to them that should trump your financial interests.
The house is presumably jointly owned. Meaning whatever happens you are each going to 50% of the equity. That’s it unless you mutually agree something else. If he wants 50% and is clear he isn’t going to change his mind then make your peace with it. A court isn’t going to give you more.
The only issue a court can look at is the welfare of the children. Under the children’s act they make an order to defer the sale of the house and decide on who occupies the house during that time. They will only do this if the circumstances warrant it.
The circumstances are that you could not afford to adequately house your children with your income and share of equity. They would consider alternative housing options, be that a smaller home or a rented home. They would expect you to maximise your income. All he has to do is produce details of local affordable homes to justify his case.
Neither of you need to go to court to determine any of this. If you do you will spend £18k between you. If a lawyer is telling you he will win then he will win. Either way you will end up wasting money on fees, redemption fees or legal fees.
If I was in his shoes I would want a sale asap and in a way didn’t lead to legal costs or early redemption costs or having to rent a place of my own for well over a year.
The law and how it is applied won’t change because you think it is absurd. All you will find is that you are charged a lot to find that out.
Have you considered what you can do to improve your income? Because if you cannot rent or buy now what will change in a year?