Of course I can't prove how high everest is, and yes, estimations have varied (and some include snow, some don't). Modern technology allows us to know its height with increasing precision. However, saying I can't know it's exact height doesn't mean I don't know it is very, very tall. Visibly taller than any other structure in the area, and well over 10km in height (I'm sure you could work this out simply by trigonometry).
Edit: Nuked a mountain eh? They did that on the quiet...the very quiet...for a nuclear bomb. If the government was in the business of nuking mountains to kill FE theory, they'd just kill you. If nuking a mountain doesn't arouse suspicion, then killing you and the whole FES is nothing surely?