What is your actual plan?
Ex takes over mortgage and ownership of the house, and moves into the house. You and your daughter move out.
Ulster Bank are under no legal or moral obligation to allow you take your name off the mortgage.
In my mind we are offering them a solution to a messy situation
Is the current situation messy for Ulster Bank? It doesn't sound like it. They don't need a solution. You do.
UB are being totally reasonable to ask him to prove that he can afford the mortgage. It's not quite the same, but offers of split mortgages are usually provisional. The mortgage holder has to prove that he can meet the repayments for 6 months.
If you are planning to move out and if your ex can afford the mortgage on his own, it doesn't make a huge difference to you. If they refuse to allow your name off the mortgage, your name will still be on the deeds. If the house is in positive equity, you could force a sale if he does not meet his repayments.
I do think that they should give you something in writing. Maybe not a commitment, but an indication.
Your ex should formally apply for a mortgage in his own name. This may prompt a refusal with a note why it's being refused.
Alternatively, you and your ex should do a side agreement that he takes full responsibility for the mortgage and you renounce your interest in the house. If he fails to get your name off the mortgage within 2 years, then the house would be sold.
Brendan