You've found out how much the clinics cost haven't you? That and how because it's private healthcare they have opinions and standards (or at least they used to) and won't help just anyone to have a child. DC about 2yr old is it? UC making noises about you returning to work...
Credit to you for not risking your health by jumping into having a ONS with anyone who'll have you, just to get pregnant. But no, don't ask your ex who is in a new relationship to father another DC with you that he doesn't want, won't want to pay for and will barely see. That's just making you look tragic and will be massively embarrassing for everyone involved, not to mention mightily pissing off ex's new partner.
Return to work, wait for DC to need you less, look for a new relationship with someone who wants DC, hope you find someone and hope it's not too late fertility wise. There aren't really any sensible shortcuts. Alternatively if you meet the correct criteria for it, maybe look into adoption if you're desperate to offer a DC a home?
There are websites where you can find men who want to be fathers for this sort of thing, that's probably a better bet than asking your married, minimally involved ex to do it.
I read about that in a magazine once, an undercover journalist looked into it. 99% of them were only interested in achieving pregnancy by "natural insemination". They weren't sperm doners in the traditional sense, they were a bunch of chancers who liked no-strings sex and didn't like condoms.