Absolutely you should petition to have the support arrangement adjusted!! If he wants to send her to private school, then in my experience, YES, it is very likely that he would have to pay for it, unless there is some specific reason that the public school is not suitable for your child.If he is concerned about how you are wasting his money or something like that, then it can be set up so that he pays the school fees directly, I believe.
My attorney is Nanette McCarthy at Griffin, McCarthy and Rice. Their offices are downtown and Nanette will listen to you and really helped me work out a thorny issue with my ex last year when my mother and I wanted to take my 10 year old son to visit relatives in the UK. Their website is: http://www.gmrfamilylaw.com/
Give Nanette a call and don't let your ex make demands that he wants you to take care of, but is unwilling to support himself! good luck!