I wouldn't get just one neon. They would do fine in the same water conditions as a betta, but I wouldn't put them together. I feed mine tropical fish flakes. Youre talking about neon tetras right? The little red, blue, and clear-ish ones?
I think you might just have a passive betta. Flaring and building nests are meant to defend their territory, maybe your male is just mellow. I have one male who is very aggressive. He flares at every plant, reflection, movement, etc. I have yet to see my other male flare or blow a single bubble, granted I just got him and he has pretty bad fin rot. I'm sure Hydro is quite happy.