Using the original tune is generally out of the question as Nintendo, as best, will demonetize it with a copyright claim.
Using a remixed version is a bit more acceptable as Nintendo doesn't own the rights to the performance (only the tune itself) & therefore make things a bit more complicated in terms of copyrights. However, as long as it's generally for non-commercial use... the creators of the remixed version will generally be OK with it.
If you choose to recreate the tune yourself, you're in the best legal situation as it's your own performance of the tune, even through Nintendo owns the copyrights to the tune itself. It's harder for the video to get demonetized because you're the one performing the tune instead of somebody else.
If you're not concerned about demonetization, you'll get by just fine as long as the music isn't the only thing in the video.