Neon tetra disease is not specific to neons. It is named after them, but just about every fish is susceptible to it.
I agree, danios and skirt tetras are not good choices for your tank. Because the tank is so small, you should choose small fish - though big enough not to get eaten. And fish that aren't very active. Aside from the size of the tank prohibiting active fish, it's better to not keep such fish with a betta.
I've kept neons with bettas without any issues whatsoever; I never found them to be nippy at all. However, I had a large group of glowlight tetras that were really nippy, yet they are not a fish people normally find to be nippy. There are a lot of factors that influence a fishs behavior.
Any time you try keeping bettas in a community there exists a risk that either the betta will nip or be nipped. Many have success keeping fish with a betta. Some don't. Sometimes something that has worked for a while all of a sudden becomes a problem. Stocks work until they don't. I've had fish together for a couple years that just up and decided to no longer coexist. It happens, which is why it's important to have a backup plan, even if it's working.
125 - BGK, chanchito cichlid, pictus cats, silver dollars, palmas bichir
125 - cichlids (severums, bolivian rams, chocolate), rainbows ( turquoise, red), loaches (angelicus, zebra, kuhli and horseface), plecos (BN, RL and clown), denison barbs, tiretrack eel, pearl gouramis, betta
90 - Congo tetras, african knife, upside down cats, spotted ctenopoma, kribensis, delhezzi bichir
2.5 - betta