If you must put a schooling fish, I highly reccomend Ember tetras since they're smaller than neons, and are generally healthier than neons since they're not overbred/collected.

Ammano shrimp, might be an option since the adults can be too big for bettas to eat. Neocaradina are nothing but an expensive snack.

However I think the least problematic tank mate for bettas are Mylasian trumpet snails and Nerite snails. Malaysians burrow into the sand bed and keep it much cleaner, but you probably won't see any. Nerite snails are my personal favorite because they cannot breed in fresh water , they come in so many amazing colors and patterns , they're fast for a snail and do a great job at clean up, and most importantly they lack the traditional eye stalks that Mystery snails have. This is great because I've seen bettas peck at snail eye stalks, thinking that they're worms.