G.B.
So he's paying half when asked. He didn't want to waste money on extra tutoring until her realized how much she needed it. Now he's paying a full half.
He wants to spend more time with his child and this is interfering with the tutoring. So he wants the tutor to come to his home for the sessions and is asking you if you'll pay of this charge.....seems like he can see the relevance but it's bit skewed from a normal point of view.
So here's what I would do. I would rearrange the schedule so that tutoring is not on dad's nights. If the tutor is working with her 4 nights per week then he gets her the other 3 if he wants. That seems fair and in the 60/40 sort of ratio. You get to keep her on the nights she has tutoring, that's 4 out of 7. Then he gets her on the non tutoring nights, 3 out of 7. Everyone's happy.