If you really need p-values, I would just use MC methods to calculate them directly. It may be less costly and error prone than what you are doing, as these methods are relatively well established.
Sampling is cheap, but probably not that cheap 
i.e. I can’t sample for each k (size of Wₖ) in 4:5000 (and I’m not sure it’s feasible to compute those covariances for k > 500), so I think I need to take some shortcuts.
I hoped to compute the normalizing transformation for a couple ks (say k=5:5:500) and then find some approximation of t~t(k) (it seems to be exponentially decaying) to be used for all k.