Mo it is unclear to me what outcome you are looking for.
Surely there is zero chance he will pay the mortgage and the child maintenance while you live in the house with the children? He is already refusing to pay the CM. It is going to be virtually impossible to force him, especially with you being unmarried and him having an overseas bolt hole.
You can't afford to keep up even half of the mortgage and bills payments yourself or buy him out can you?
As long as you are in that house relying on his goodwill you are screwed.
I know you don't want to sell up and move. Maybe I'm being dim but what other option is there that genuinely exists?
Your reasons for not moving are a bit lame (sorry). Your DC need you happy. That's way more important than living in a naice house for every single day of their childhood.
Have you had a serious look at what you could rent for a year while you find your feet, get a decent job and eventually become eligible for a mortgage yourself?
Selling is a thing you can force and it only needs him to be a tiny bit reasonable and only for a short time. When you've got that lump sum in the bank, he loses most of his power over you. You'll even be able to afford a lawyer and full page newspaper ads to get the damn CM out of him.