I, too, was coming on to mention the word “allowed” cropping up a few times, but have seen you’ve responded to the pp about that now.
However, despite you not being religious, it is also your daughter who is getting baptised, and as such you have every right to invite whoever you want to. Especially as your father is paying! The fact that your partner is issuing all these orders and rules as to who can and cannot come would piss me off immensely, and I would completely ignore him! That alone would have me telling him that he should bugger off with his “allowed” list and that you will be inviting your family - and you mention that some of your family are Christians anyway? But the fact that your father is paying just takes his shitty behaviour, regarding this, to an entirely whole new level and absolutely no way (in hell!! 😈) would I be doing what he says. Invite your family, the gathering afterwards is also important, christenings are meant to be a celebratory day and having wider family there as well is so important (those who you want there and can attend), because this day is about your daughter and the more - close - family she has that love her and want to be there the better!
Honestly, let him sulk and get over it, tell him they are coming. Don’t give in and tell him he will have to uninvite them himself, just tell them they will be there, like it or lump it, and if anything is going to ruin the day it will be him if he chooses to be so petulant about it!
I still can’t get over him thinking he’s the law on this, especially as your father is paying! Wtf is he thinking that he even thinks it is only up to him?! Put your foot down. It’s your daughter and your family so tell him he should be happy that so many people are there for their niece/cousin/whatever ( aside from just the immediate family) and that she has so many people who love her. And that it is going to happen, whether he agrees or not. His cousins aren’t immediate family either so tell him to suck it up. Or pay for the whole thing himself if he thinks that what he says goes.
eta: to clarify when I mentioned inviting who you like, and the after gathering in the same sentence up there, I didn’t mean invite them only to the party part, I still meant the whole day.