The question is "can they prevent me from going with the school/class if I refuse to provide a background check?" There are many factors that coudl affect the decision as to this issue. Soem are, the mode of transit, the interaction with other kids, etc. You do nto have enough space here to provide all the information to utimately determine wif you coudl sue to force the school to allo wyou to go without a background check.
On the other hand, if the school/class is going to the zoo, they cannot prevent you from going to the zoo too. You just may not be able to travel with them or participate in their sanctioned activities.
Good luck
