The only way this would be possible would be to upload them onto that in-game thing, save it under someone elses video, and then you can do whatever to the old one. Basically:
1. Go the the GTS and upload your battle.
2. Take that same battle and save it under Someone Else's Battle.
3. Do another battle, save it, and then you'll have 2 videos, one listed in your own battles, one listed under someone elses battle.