Why no regens? It's not like that fights are gonna last too long. And if the fight does take too long, you can simply prevent it this way; make it a rule that fights like these can't last longer than 10-15 minutes. After that period, in case they didn't kill each other, the team who did the most damage -- wins. Just a rough idea.
I've already kudosed you several times for efforts you display in organizing your tournaments. I'm also a big fan of your comments during battles. Can't wait to participate in that cruiser tournament:)