Low pass filtering effect on accuracy of power measurement
If I'm reading your scope correctly the fundamental frequency of that hash is about 2kHz, and it doesn't have a lot of sharp edges.  That suggests that you've got frequency content up to maybe 10kHz at most.  Depending on how aggressively you want to filter, you could probably get away with less than 100kHz, but 100kHz (and an appropriate anti-aliasing filter) would do it.

To get the most accurate power reading you want to sample voltage and current simultaneously, multiply, and average.

This article is really directed more toward debunking common misconceptions about sampling, but should have some guidance about how to select an anti-aliasing filter to go with that signal and a 100kHz sampling rate.