Rarely will admin take a side over a student not being able to do both activities if logistically feasible.
Also the parents of those affected can go to the principal as well to bat for being involved in both programs.
Your program should take precedence (without being narcissitic) as these are scheduled activities and are normally UIL or tangentially UIL-related. The NJHS events are scheduled at will generally and not coordinated with other schools necessarily. You may also talk to your administrator about who gets on the school calendar and why those dates take precedence.