I think you answered your own question. Here is my opinion
Don't do one together. Then let dates decide who does it.
Tell her that her dad should organise his time with kid when she has them. Otherwise it is your time. You can make an exception this time provided you get the time back.
This is not about being an ass nor even about not being a doormat, though that is important. This! Is about you doing what you want and respecting yourself and your rights.
R 25 years M 14 years S11 & S13 Working on it alone since Oct 2014 M in trouble a lot earlier (~2 years) Feb 2016. 1st R chat in a yr. Next R chat Aug'17 Still together