Im sorry to hear that your hurting compassion14
As for his behavior with as dysregulated and unpredictable our exs can be its hard to say how he will react to contact regarding his financial obligation. On one hand BPDs hate responsibility. When my exBPDgf used to talk of us getting an apartment she just wouldnt have it to my asking her to be on the lease. "It was my responsibility as the man and she didnt want to be stuck." So what do i do get an apartment for us and she "needs a break".(aka cheating on me) My point being they tend to stray from any obligations and usually are very avoidant. On the other hand he could actually help if he has something to gain out of it from you. But since you said hes likely idealizing other women id say the latter is more likely unfortunately.
Trust me i get it. Its very hurtful how they stab us in the back then rub salt in our wounds. Mine owes me a huge sum of money that i loaned to her for her car and cell phone plan. In time ive learned to accept as unfair as it is that ill probably never see that money again especially now that shes using heroin/opiates again.
How do I handle him moving on and how best do I handle the situation prior to thrashing out the financial stuff just now when I really don't want to be painted black as he idealises someone new and discards me.
In regards to him moving on theres not much you can do to stop this. He'll either come back around or he wont but either scenario will ultimately lead to you being painted black at some point sadly.They can only see in black and white with their splitting defense mechanism. Dont take it personally its the disorder not you and the disorder always wins. You can disengage and go NC to handle his moving on. But LC seems to be necessary to at least until you get this matter sorted out. Now as for this shared ownership is their any legal recourse you could take from him signing any liability towards his share?