Anyone know a good method for putting videos into stamps?

I tried multiple ways to do it, but I ended up having to crop the looping video and re-adding the stamp frame to an entire image sequence tediously and them tying it back together into a gif... What kinds of software and methods do you guys recommend? This is my 2nd stamp gif I made. (The video is the looping fire background.)

(I don't really expect anyone to understand the joke from this one... This is Carp Boy, the mascot for Hiroshima's baseball team. The fans in Hiroshima and Yamaguchi are crazy for this guy, and there is so much merch for him. So I made this stamp based on my experience in Japan. You can nevertheless use this if you want lol)