A bigger tank will be great for him, and probably for you too because it will cut down on a little bit of tank maintenance for you. (and give him a big area to explore and swim around in! Somewhere I read a rough estimate of a normal territory is about a 5 gallon area... I don't remember where I read that anymore or if i'm mistaken, it's been a while.)
As far as the filter goes, as long as the current from the filter doesn't blow him around a filter is good for the water quality. Almost any filter can be modified to be betta friendly though with additions of a baffle and a sponge or nylon (etc) on the bottom of an intake tube.
I'm a big fan of Hydro sponge filters, but they are unattractive to a lot of people visually. If you are interested you can probably find information on the Hydro, and other brands of sponge filters by tossing up a google search or something. They don't create a strong current in the tank.
Edit: For the damage on his fins, think that the disease forum would be best.