Egh I'm the ex wife in this situation and honestly I think your entitlement has trickled down hill.
You move your dd to another country, a country that has state free schools, but you don't want that so ask for him to pay for half of a boarding school which is something you wouldn't be asking if you hadn't moved.
And frame it as how dare he book a wedding when he should be funding half of the private education you chose in a country that you chose for your joint dd. You aren't his boss and actions have consequences, you chose to move to another country why should he shell out for private education when there are state funded schools.
Your daughter is young, and you have set her up for the pick me dance with her own dad. That's so fucking grim.
I think it's cheeky for them asking you to pay for your dd flight for the wedding, I think it's wholly unreasonable that you think they should pay for the your dd new bf flights and additional cost of the wedding.
Also you seem to be placing the blame on him for his decline in his relationship with your dd, when you moved her away to another country and I don't doubt made your feelings known about the new baby and that your ex shouldn't be getting married and should be funding your choices you made for dd. I don't think that actually that any of you come out smelling of flowers. But that breakdown isn't just on him, it's on you and if you don't start taking some responsibility your daughter will piece together the pieces
I know how hard this type of situation is because I have worked dammed hard not to be a mum like this. And my ex married the OW who broke up our marriage when our first child was born still.
I would be highly disappointed if my dd acted like this tbh.
Oh and he's not his gf she's is fiancé soon to be wife, and "new baby" your dd half sibling. Language matters and right now yours is barely containing your entitlement. However I'm glad you didn't text and make a fool of yourself.
Blaming your exs new wife for the reason your ex is making the choices he's making is such a lazy argument. He seems to have a spine - it's just he doesn't agree with you.
Clearly.