I got in touch with a spiny mouse expert 2 days ago, and he advised me to leave them be at the moment and see if things calmed down if they were just tail biting. This morning, there was even more biting and there was blood everywhere, so I contacted him again, and he said it was probably time to separate them. So now they're in their own tanks and they actually look a ton happier!
I got very lucky because they are incredibly people friendly. I didn't want to bother them too much since they were very stressed from their move the last few days, but today, I finally was picking them up and handling them and they were super sweet! Not a single bite, even from the "bully". I hope they will do ok on their own, but I might put a tank divider in so they can still see and smell each other. I just need a larger tank first.