Antigens are actually pretty accurate, but when they are wrong it is almost always a false negative- it is rare to get a false positive. In a study of 900,000+ antigen tests given in workplaces, 0.05% turned out to be false positives. So, his chances of being sent home for a false positive are infinitesimally small. If he is vaxxed & boosted, his chances of being sent home with an actual positive result are small.
But: the chances are not zero, so you have to weigh the relative value for you - and for him- of his having the privilege of going to this camp, versus the possibility of losing a meaningful amount of money.
One of our collegekids has just had a trip cancelled altogether b/c the college felt it was just too risky. She won’t get that chance again- it’s her last year. So many kids have missed great opportunities .