I have kept neons with bettas numerous times before. The key is too keep the school of neons larger (minimum of 5) since they school tightly the betta is generally less likely to attack them.
Dont get me wrong, some Bettas are just meanies. It is always a case by case basis with Bettas...but I have had many over the years housed with neons, ADFs, snails (recently), rasboras, and other less finned calm fish without issues.
I agree with AuntKymmie...put the betta and snail in the QT and let the neons have the big tank for a few weeks. And if you do it that way, I would add 2 more neons to that 3. With small schools they get stressed and it can weaken their immune system.
Not sure if it is completely needed to keep the neons and such QTed away from the snail. I am new to snails, but I thought I had read (could be wrong here!!) that they dont carry diseases common to fish?? I dont know, but if they dont, then you would only have to move the betta.