It's been said elsewhere and above, but the absolute bare minimum evidence for this endeavor would be GPS data off of a running watch. This is 2016 -- I can see second-by-second evidence of my friends' easy runs on Strava -- so how is it that we're not going to be able to see evidence of a transcontinental world record?
The absence of this evidence is evidence of something.
There is no way to get true evidence. To make a monumental claim, the burden of proof is upon them. All a GPS shows is that someone (perhaps other than RY) ran (or transported) the GPS for whatever distance at whatever pace. The best way to earn trust of doubters, is, [shocker] to earn their trust, by
a) not making bogus record claims like longest run w/o sleep
b) bringing as much evidence to the table as possible
c) not be combative with those asking innocent and reasonable questions
d) show up to a proper race and put up a decent performance. There are many multi-day races around the world where he can prove his mettle should he choose to do so.