I agree the tank size is fine. 1 gallon is really minimum, and that's mainly for younger fish.
A full size adult betta will grow to fill up a 1 gallon and look really BIG for the tank eventually. Generally if they start to look big for whatever you're keeping them in it's time to upgrade.
No matter what you do keep the tank you have, if for nothing else, a hospital tank.
The red gills point to poisoning probably from the conditioner. I have never used that conditioner, but I stay away from anything that says "fizzy tab" or water conditioners marketed directly to bettas. (personal choice)
Prime is a good conditioner because it also binds ammonia for a time making it less toxic to fish if I remember correctly. When getting a conditioner look for removes chlorine, chlorimine, and heavy metals. (that ammonia binding is nice too.) I am using Top Fin Tap Water dechlorinator. I plan on getting a bottle of prime once this is gone. I've never had a problem with the top fin stuff though, and it comes in a tiny inexpensive bottle that treats 600 gallons.
The water companies are switching over to chlorimine now i'm sure for the summer. If your conditioner does not remove this, it could be what shocked him. (From what I understand water is chlorinated in winter, and chlorimine is added in summer because it does not evaporate from the water.
I DO use Aqueon Betta Water Renewal which just is a bottle of mineral supliments I put in a few drops per water change, not as much as they say to put in. Offhand I don't know if this stuff works but 6 drops per gallon of water has never harmed my fish.
If you want to age water you can get couple gallon water jugs of drinking water from the grocery store. Drink them and save the water jugs to fill for your fish. Put the drops of tap water conditioner directly in the jug, shake it up and set it near your bettas tank. The water temperature should be the same as long as they aren't in a draft or on a cold floor.
Here is another heater I personally like, it's not adjustable though, but I have never had a problem with them and I own quite a few.
They keep the water stable at 78 and if your water gets above 78 it shuts off until the temperature drops below the safe zone. They work in tanks as small as a half gallon safely. It's better to invest in an adjustable thermometer though in the long run. If your fish ever get sick you may need to raise the tempature above 78 to treat them, and many here swear you should keep your bettas in water temperatures below 80 which is really ideal.
I love a lot of Aqueon products. They have super customer service too if you ever need to contact them. I would stay away from that conditioner in the future though.
To everyone else talking about PH and cooking:
Gads I never knew that the ph could be what is making me mess some recipes up. I even was an avid watcher of the food network when I had cable tv... Even in cooking magizines I have gotten it was NEVER talked about...
Sorry this was long, and i'm very sorry you lost your fish. Don't let it discourage you from trying again.