I'd say that if she's calling you "Fish PETA", she's a lost cause. She's making fun of you for caring about the welfare of an animal, and any steps you take to try and better the welfare of the animal are just going to reinforce that. Some people simply will not see fish as anything more than decorations. It's unfortunate, especially when they bring that perception into the workplace, because others are then forced to witness the cruel conditions that they keep their fish in.
But when it comes down to it, animal cruelty laws don't generally extend to hobby fish, and there's not likely to be a company policy prohibiting her actions. The best thing you can do is try to ignore it and move on. You can't control what happens to the fish, but you can control your working relationships with your co-workers, and you'll need to keep those intact long after that betta is dead and gone, and she's onto a new pretty trinket for her desk.