I suspect that OP wants/needs arguments to shut him down as much as anything.
As a single, childless reasonable person, my answer is fuck no, dump this guy off a cliff, block him.
I would bet dollars to doughnuts that there is an abusive relationship waiting for you if you do this. "Quality Time" will turn into you taking a lot of shit while he does as he pleases. You're already by yourself, in a new town without close family and long term friends. Nice and easy to isolate.
He's just got to get rid of your daughter first. But he's got to get YOU to do it - he needs you to petrol bomb that bridge so she will hate you and never speak to you again.
If your ex was the right person to parent full time, you wouldn't have moved and would already be sharing custody. So there's that.
Financially, if she did live with your ex, you would still be responsible for her money-wise, and being a good mum, you'd want to step up.... So his argument there falls flat.
She's 14. That's a vulnerable age. A child might be physically capable of surviving at that age, but emotionally, she needs strong, steady and loving parenting. Weigh that against a relationship that - if I am wrong about his motives - might crumble within months. Your relationship with your daughter would be burnt toast and you have nothing to show for it (not that I can see there is anything really worth toasting a child over). You would probably estrange your daughter and then you miss the graduation, wedding, grandchildren moment s and more. For what?
There are loads and loads of women out there who would be happy to have a relationship. Why hasn't he gone for them?. Why has he not cut his losses and looked elsewhere, not come back and had another go? That gets my spider senses tingling.
Finally, your DD is 14. You don't have long before she goes off and does her own thing. This is the time to prep her for life, including showing by example how to parent. You're the best person to do that.
Sorry for the long post.