I have a betta in a 5.5 gallon tank. Your 2 gallon tank is fine, but in order to do water changes only once a week, you need a biological filter -- the "good" bacteria. I do water changes in my 5.5 gallon once a week. You will still have to measure your ammonia level once or twice a week and see if the once a week water changes are keeping the ammonia down. Larger tanks are easier to keep stable, so you still have to watch the ammonia levels in your small tank, but getting a biological filter in there will help. Even though bettas can supposedly tolerate some ammonia in the water since they can breathe oxygen at the surface, you still should keep the water level ammonia at 0. When I got my second betta, his gills got burned from high ammonia levels (he was blue and his gill area turned a brownish color) and this area never turned blue again, but he lived to over a year old. So, it's important to keep ammonia levels at 0 or as close to 0 as possible. When the tank cycles, the ammonia reading will be 0 consistently (unless you add another fish, which you probably won't to this tank).
Do you have access to Bio-Spira? It's about $10.00 for a one ounce pouch, and you're not going to use the whole pouch, but that would be the best way to get your good bacteria in there for your biological filtration. If you add the Bio-Spira, DO NOT add Amquel. The Amquel will end up binding to the good bacteria in the Bio-Spira. The good bacteria will not get your biological filtration started in the presence of Amquel, and then the Bio-Spira will be wasted.
The other equipment I have in my tank is a Jungle Jr. Dirt Magnet filter, hooked to a TetraTec Deep Water 12 airpump (but it's a very small, quiet pump) and also run through a gang valve.
I don't think a
UGF would be helpful. It's such a small tank that I dont think you could find one that fits. I have never had one, but I think you need a good airflow/ water flow to go through them, and that may be too much current for the betta.
