IMO/E-in the proper set up with healthy fish...the betta and neon can coexist without issues, if the tank is too small or the neon is weak or sick the betta can kill it, but in the proper size tank and healthy neons....no way can a betta catch them and kill them.
Neons are fairly large fish and I don't think even a king betta mouth would be big enough to swallow a grown neon tetra, I have kept neon and betta together for years and have never lost one from a betta killing it.
I feed my betta guppy fry and once the fry get to 3 weeks old they are too big and fast for the male betta to catch and swallow, the female can still catch them and tear them up until about 5 weeks.