Before you introduced your betta to the new tank, did you cycle the tank? From my experience, fin rot usually happens when an aquarium's parameters are poor or unstable leading to stress in the fish. This stress can lower his immune system and leave him susceptible to diseases and infections like fin rot. Keeping the water stable is easier in a bigger tank, I would highly encourage you to consider getting a bigger tank (5 gallons absolute minimum) as I personally believe 1 gallon is far too small for him to live happily. Also be careful with 100% water changes as the sudden change in water can stress him even further. Also since you were keeping him in a small tank, 2 week water changes is way too infrequent. Depending on your water parameters, I would probably be changing out 20% of the water every 2 or 3 days.
As to what you can do now is keep monitoring him and the water parameters. Keep the temperature stable at around 75-80 degrees Fahrenheit . Fin rot is absolutely treatable. Ammonia and nitrites are highly dangerous to any fish so if you detect any on the API strips, it's time for a partial water change (not 100%). Nitrates are safer and only get dangerous when in high concentrations of 70ppm. If you can get your hands on Indian Almond leaves, that would be great. It's a leaf that provides natural anti bacterial and fungal properties and has helped my bettas. You can use those with the medication. Just boil it for about 10-15 minutes, let it cool down to match the temperature of the tank and treat it with water conditioner before adding it to the tank when doing a water change. It'll turn the water brown but that's normal. Also when using medications, be sure your filter doesn't have any activated carbon as that removes medications in your aquarium.
