Like you added in a comment, a plant like java moss or something that floats would probably make the fish more comfortable. Snails would also be acceptable as they are interesting for curious or bored fishes.
Neons start to bite each other's fins when they are stressed. Being in a school makes them less stressed, but only 2 fishes is not enough for this effect, you would need at least 6 to make them feel safe. Actually they might bite eachother more, being 2 or 3 stuck in a small space. The other fishes don't need medical treatment, so it's not fair for them, and it could stress them for no reason, even if there is no disease here, only fin damage.
I understand that you think about the social needs of the Neon, but my opinion is that your sick/hurt fish should be isolated, and we know what "isolated" means. If it was sick with disease, you would never mix other fishes with it. In these cases the healing process is more important that the schooling needs of the fish.
Your setup seems good as is, since your breeder tank floats in the main, the fish is already sharing water with its friends and can visually see them, so that's reassuring enough. Focus on not stressing this fish too much and keeping him in good conditions, and he should heal. If the breeder tank is floating, please consider if it is too close or not to the aquarium lights, because your fish might be unable to get away from the blinding or heating of the lamps and that is stressful.
If your fish gets healthier, you can allow temporary test visits in the main tank, and put him back in the breeder tank later. (if you are worried it is lonely or lethargic, this would help for sure)
Thanks for taking special care of your hurt Neon.
Also by curiosity, how big is your main tank? You should take note that 13 (neons) is an unlucky number, in the future try to deal more with 12 (very versatile in mathematics because it's easily divisible) or with 14 (it's lucky because it is 7 + 7). Multiples of 5 are also reliable numbers. If you don't play at least a bit with numerology, numerology will play with you ! Good that you have the betta at least to bring the total to 14, but 14+1 would be more safe ! :)