Wel, in all honesty, you both need to sit down like adults and work out how you can make things better for your child. The animosity you feel towards him and his actions are likely to be playing a large part in her negativity to going.
I see the fact she doesn't want to spend time with her father as a much bigger problem than any 5yo birthday party tbh (they might seem über important now, but really, in the grand scheme of things, children do survive without attending them, ever).
So, I think you need to faking it, losing the negativity, and bigging up the time she is spending with her dad with her, and the two of you adults need to sit down and discuss dance and gymnastics, and whether it is possible, with a joint custody situation, for her to attend them. At 5, this stuff is easy peasy - when she is 11, it's going to be a whole lot worse - way more training, and way more competitions, more travelling, and a lot more time spent. I would also add it is going to be impossible for her to continue with dance, gym and swimming at a comptetitive level. Completely impossible. All three eat time, and conflict with each other and other real life.
You both need to be on the same page with this. And you need to be on the same page from now. What did he say when you first discussed her weekend hobbies with him that would impact on his time with her? I only ask, because I have several friends in similar situations who made sure to discuss their daughters dance classes with the their ex prior to registration - one family now has a daughter in dance (and in fact, her father sends an additional monthly cheque to pay for it), and the other family do not have a daughter in dance, because the child spends two full weekends a month with her father - she is a little older, and the school attends five competitions a year, with additional practice every Sunday lunchtime from 12-5 in the four months leading up to comp season.
At 5, your priority really shouldn't be maintaining your daughter's schedule. It should be how you are going to rebuild her relationship with her father. Worry about the hobbies later, once this is fixed.
Sure, it may not be possible. She may already believe that dance and parties are more important. But you do need to try.