Double dosing will not damage the fish. Half a gallon is not very much room for your fish, though... :(
There is likely no difference between the betta conditioner and the regular conditioner. Read the descriptions, what it claims to remove from the water, and the major ingredients. They will probably be the same. There is a whole line of products out there "specifically for bettas" that are a crock of s**t. Bettas need room to swim, clean water, and meaty food. Anything that claims to be the best for bettas is likely a lie, because bettas do best in 5+ gallon heated, filtered, HEAVILY planted tanks. Anything else is a marketing gimmick. This was not meant to criticize your betta care or tank size, just a heads up on bettas. There is a lot of misinformation out there, and unfortunately most people (myself included) learned this the hard way, or after money had already been spent on useless betta stuff.