In post #0, the OP said that the student chose BC, but the OP (parent) wants to hold him back.
Since he is three grade levels ahead in math, it seems like you are underestimating how good he is in math, at least for high school math up to and including calculus BC.
Most students properly placed that far ahead in math find all high school math including calculus BC to be easy A courses. Slowing him down to AB is unlikely to significantly reduce the amount of time spent (there will probably be similar volumes of homework, but more on each topic over fewer topics). The exception would be if he was pushed ahead beyond his actual math skill and subsequently struggled in algebra, geometry, and/or precalculus, but you have not mentioned that being an issue.