"If you bring a small child, please keep him/her with you so it does not distract the student performers."This solution is the worst of both worlds: clunky, and inaccurate.
We know they were trying not to raise the dander of the gender police. There is a perfectly accurate, non-offensive way to do this, however: Use the plural.
"If you bring small children, please keep them with you so they do not distract the student performers."