He seems to have swim bladder disease or a bad case of internal parasites. Half a gallon may be OK or cruel depending on who you ask. However 1 water change a week is definately too little, even if the water change is 100%, you would probably need a water change evry other day maybe at 50% and 100% water change done atleast once a week. Tap wate is just as good as bottled water when you used a water conditioner, if the bottled water is filtered then it is actually bad for the fish. Fish needs some trace elements thay can be found in tap and bottled non filtered spring water. If your temperature is cold (less than 75 degrees or less than 80 degrees, again depending on who you ask) then the fish becomes very susceptible to disease. If he is not moving at all even when you push him gently, then unfortunately he has passed. Read up a bit on bettas in this forum, ask more question about the conditions your other fish is in and hopefully your other betta will live a long healthy life.
Oh and definately super clean ur hands if you physically touched your fish. Fish sometimes carries bad stuff transmittable to humans.
If hes dead, dont flush him, give him a proper burial or wrap him up and put him in the trash. Flushing the may introduce bad things (forgot what) into your local waterways