White's approach estimates the dependence structure (simply speaking, the variance-covariance matrix) of the test statistics (such as t-values or p-values) that are used for deciding what strategy is superior. The estimation of the dependence structure is not explicit, but that's what happens during bootstrap or simulation. Typically, the amount of data is not enough to estimate it (think of how much data is required to estimate 1000*1000 variance-covariance matrix reliably) but as far as I know that problem is swept under the carpet in all of the papers of White and his followers.
BH (1995) assumes that the p-values are iid U(0, 1) under the null hypothesis, with a couple of technical caveats that are not going to be of help to you. Then, they also followed the idea of White, i.e. tried working with dependent p-values via resampling in this paper
Yekutieli, D., Benjamini, Y., 1999. Resampling-based false discovery rate
controlling multiple test procedures for correlated test statistics. Journal of
Statistical planning and inference, 1999
but the limitations of this approach are exactly the same as those of White.
I would suggest using Efron's Fdr/fdr because, in a certain sense, it provides a viable solution when the test statistics are dependent, among other things. You can start with this paper, and the corresponding book can be found on Amazon. The locfdr package is no longer available in R, but I just downloaded and installed it separately.
All of that being said, the ultimate test is seeing how the "best" strategy works out of sample.