Yes, tetras can be very nippy and not good tankmates for bettas. I would not put him back in with the tetras, and if you can, keep him in this 15 gallon tank. He could have a few tankmates like ghost shrimp or cory cats. (cory cats like to be kept in groups of 3 or more)
But let's get him well first -- the stable temp. at 80 is good, and room light only is fine. He's probably eating and swimming more in this new tank because the tetras aren't ganging up on him, so he is not stressed anymore. If his colors are getting better, that's another sign that he is no longer stressed.
Dont' use only distilled water for changes -- it doesn't have any minerals in it that fish need. If your water is very hard with a high pH, though, you could mix the tap water with the distilled water. My water out of the tap is pH 8 or 8.2. In an empty, bottled water gallon jug, I mix half tap water and half
RO water. My pH (with the mixed water) is 7.2 or 7.4. This seems to suit the bettas just fine.
I'm not familiar with betta plus conditioner. I use Novaqua and Amquel. Just use a conditioner that removes chlorine and chloramines. Some chloramine removing products like Amquel remove ammonia too. As long as you don't use more than 8 or so drops per gallon you'll be ok. You don't want to remove too much ammonia because that will interfere with the nitrogen cycle. This new 15 gallon tank, if it wasn't cycled before the betta was added, will go thru the nitrogen cycle. Read about it here if you're not familiar with it.
http://www.aquariumadvice.com/showquestion.php?faq=2&fldAuto=21
I just googled the Betta Plus conditioner. There was no real description other than it said it was a betta bowl conditioner. So you may use alot of that little bottle for 15 gallons. It would probably be ok, though. I prefer Novaqua and Amquel because I know that they don't interfere with a cycling tank, when used as the directions say.