I'm a stepmum to two and we also share a child. I've always been slightly on the fence about what I should and shouldn't pay toward his kids. For me, splitting household bills when we live together 50:50 is just what we do. I don't make him pay 70% because he has two more children. Admittedly as they've gotten older the food bill has rose significantly (teen boys who hoover up anything in sight!) So DH does buy more of their treat or snacky stuff separately from the main food shop and he pays for that himself as it just gets demolished 5 mins after the shop is put away.
But anything like the mortgage, council tax, water, gas and electricity... is paid 50:50. I just cba working out who pays for what tbh especially since we have a child together anyway.
That being said I absolutely do not expect to have to buy anything for them personally I.e. their clothes, contribute towards maintenance, uniform, school trips, games etc.. their dad and mum pay for all of that just like me and DH pay for that for our child together. I don't think it requires a 3rd person's contribution, I mean I don't get some 3rd person buying all that for our kid so I'm not about to do so either for theirs.
Obviously there are times when that's not a strict rule I.e. I may buy them the odd thing every now and then if I'm out and see something or at Christmas for example, but I'd never accept it being expected that I contribute to DH and exes kids in that way. Same with holidays tbh, DH pays more to account for the fact he has more children (who are also older and therefore more expensive as no child space like ours).
So yeah I think your partner is shitty for not just accepting bills will be 50:50. But I think anything outside of that to do with your child is for you and his dad, nursery fees, clothes, trips, uniforms etc..