TL;DR: Film the whole trip in one shot, back and forth, and publish the video online.
Also, during the trip, film a screen with the latest Bitcoin blocks, the last tweets publicly written and/or latest BBC news for timestamping. You can use any other source of public timestamping of course, like the feed of stock prices, etc.
(Optional: Upon retrieving the flag, shoot it with a paintball (indoor, against a hard surface) and film that. Making the exact same strain should be impossible for anyone else because its exact shape is random, especially on the edges of the strain.
This will ease future authentification, if the flag is lost and found again but the video is proof enough that you're the one who brough it back to Earth.)
As soon as you are back on earth, use the lastest Bitcoin block or any other public timestamped source of randomness and hash the video along with this. Send it to Nasa as soon as it's done so they know you didn't have time to forge the video.
Shooting the flag assumes you are ok with straining the flag
The NASA will see that a spaceship going back and forth, that the flag is missing, and you provide proof that it is in fact you who retrieved it.
You can even livestream the event or publish the video online right after landing to skip the paintball part.
Bonus: you can joke about the US or the NASA, or express your political views, and completely control the narrative for extra humiliation points.
Edit: What if you want to release proof at a later date?
Do the exact same thing, except that instead of releasing the video right after it's shooting, you make a cryptographic hash of it, which guarantees it's integrity, and publish that instead. When you decide to release the proof, release the unedited video, and geeks worldwide will confirm it's the original.