Bettafix does contain Melueca (sp?) which is a form of tea tree oil, which can coat the Labyrinth lung on the Betta, although the company insists that deaths resulting from the use of Bettafix are because the fish either didn't need the bettafix or it was used incorrectly. They maintain that bettafix is not harmful if used correctly and only used for what it's supposed to be used for. (I emailed the company and got a very lengthy reply on this)
So, to be on the safe side, I would not use it to repair his fins, especially since it sounds like it's fin biting and not fin rot. Warm, clean water (frequent water changes), are the best thing to heal this.