Yes it is. Enroll the kids there or not. You can't and shouldn't be able to have it both ways. Having it both ways would be most unfair to people like me that have chosen to send kids to private schools. I still pay school taxes. I don't get a thing from my school taxes. However, I wouldn't even think about trying to get my kiddo into the local middle school band because her private school doesn't have a band. My daughter doesn't go to school there. The principal would be fully within his/her rights and correct to tell me my daughter can't play in their band because she doesn't go to school there. It's pretty much a no-brainer.
You should be able to. It is all to the good when the student takes a place in the commerce of the community.
If your private school doesn't have a band then your kid should be allowed to try out for the PUBLIC school band too. Your problem would be that you wouldn't be able to take your child out of the other school at a particular time because he would have another class to attend. Homeschooler's don't have that problem.