There are two issues here which you seem to be conflating.
1) His joint ownership of the house
2) His joint liability for the mortgage
Your solicitor will need to convey his share of the house to you so that his name comes off the deeds and he no longer has any legal ownership.
The lender will have to agree that you can afford the mortgage on your own to give you a new mortgage in your name only.
Getting his name off the deeds is the first priority.
Don't worry too much about your bank. You can get a mortgage elsewhere if they refuse.
However, I understand that a lender won't give a loan to a married person on their own.
So it does seem that you need to be formally separated.
But the first priority is to get his legally binding agreement to convey ownership of his share of the house to you.
If it takes time to close this sale or change the mortgage, so be it.
so far seems to be going OK.
I don't know how often I have seen this and I have always given the same advice. Reach agreement while you are still getting on with each other.
All it takes is for one person to see or hear that the other has a new boyfriend or girlfriend for things to suddenly change.
Brendan