Good news: Managed to find the cup he came in and QTed him in it, gradually mixing the tiny bit of tank water I caught him in with mostly water from my other, healthy bettas tank (so conditioned and cycled water, but with no melafix), this way I can monitor his poop like callistra suggested and while he's getting his salt treatment (I put a few drops of salted water in his QT cup, I'll add more tomorrow until I get up to the full dose) I can see if the neon's tails heal without him in the tank.
Bad news: He's looking more bloaty again since the last time I checked on him even though I haven't fed him :(
Really can't thank all of you enough for the advice :)