Thanks @KjetilBHalvorsen. I think FDR will be very helpful. That said, I think I've convinced myself that testing p-values is OK, assuming the p-values came from a "correct" test. In this case, by correct I mean that over all inputs, the test produces p-values that are uniformly distributed. With a random input, the test effectively becomes a random number generator on [0,1]. This should qualify it for the AD test. (Of course, taking your point that the chi-squared test in particular is approximate).