If we were married a court would say that he should move out, and yet because we're not, they won't.
I'm sorry this is going to be another brutal post op, but it sounds like you need a reality check.
A court would most certainly NOT say that he should move out. You would certainly be expected to be able to afford to buy him out ENTIRELY, AND be able to afford to pay ALL the bills by yourself, even before negotiations began about equity shares.
Even when married you don't just get given the house.
Assuming your early repayment fee is currently £18k in the penultimate year, does that mean your mortgage is astronomical (2% erf means your mortgage is £900£k)?
All you are legally entitled to is 50% unless you set it up another way. You will be able to house your children with any equity released by the sale, then would be expected to maximise your income (as a divorcee would also be) to fund your bills. Plus whatever child maintenance is owed.
The sooner you accept this the better (and cheaper) it will be for you.