I don't know if Brita filters remove chlorine, but water conditioner also removes heavy metals and things that are bad for fish, it also helps build up their natural slime coat. My tap water doesn't have any chlorine in it, (Well water, I live in the country) but I still use water conditioner, because it removes more then just chlorine. :)
Yes, bettas can and do die because of extreme temperature changes! I had two boys in a divided tank and I forgot to plug their heater back in after a water change (NEVER do water changes when your really tired!!) and the next morning, they were alllllmost dead. Luckily I managed to save them. But one of them has been battling fin rot ever since.
You probably want a tank bigger then one gallon any way, get a 2g+ tank and a heater, and your little buddy should be happy! You can get a small bottle of water conditioner for a few bucks, a few drops per gallon. LOL I have to get the huge bottles because I have six tanks. One bottle wouldn't even last one water change day. ;)
Good luck!! Defiantly get another little guy or girl! They are very very fun and rewarding!