I've never had a tail biter, so he could certainly be biting his tail. My only experience with a frayed tail is my double-tail. He was frayed due to the increase in swimming space. If your guy went from 1/2 gal to 2 gals, he's been swimming a lot more than he's used to. It's like a flag waving in the wind - after awhile it gets frayed. Another possibility could be your filter intake. Is it baffled? He could be getting his tail caught in it. Just some other thoughts to consider. I can't really give advice on the best medical action to take, but I've read on this forum that super clean water can prevent any infections with frayed/ripped tails. He doesn't appear to have any bacterial issues so far. AQ salt assists with the slime coat, as well as disease prevention, but if you're strapped for cash, then the best prevention is super clean water.
He has no problem with the filter, never been sucked in or caught. In fact, he sleeps under it and likes floating next to it. I clean his water once every two weeks since there is the filter and it seems to stay clean. I hadn't thought of the increased swim space causing the damage. But, i suppose it is a possibility.
My big crowntail shows no signs of frayed fins, but my double tail looks like he's been swimming in a wind storm! :) If you have a 2 gal, even with a filter you'll need to do more frequent water changes. Any tank under 5gal is very unstable with any type of cycle, and while a filter is great at removing particulate matter, it won't remove the excess ammonia and harmful nitrites. I checked my 2.5 gal after only 4 days and my single female had produced almost 1.0ppm ammonia. Anything over .50ppm can cause permenant gill damage and any nitrites over 0 immediately starts effecting the oxygen flow to the blood. Those chemicals are bad stuff. Especially if your little guy is showing signs of fin damage, you'll need to up the water changes to insure he doesn't come up with a bacterial infection.
I have been thinking about it and i was reading that bettas like to have plants and things to hid in. I have one plant in the tank and it isn't very big. Also, I checked into stress coat and found it is similar to the water conditioner i am using....What i am thinking is that he might be bored. Since they like it to be alittle darker and have places to hid, I was thinking i might buy a cave type thing he could hid in and possibly get a small mirror that he can flare at for a few minutes a day
boredom, stress, or just having too much finnage can be all reasons for tail biting. I think adding the decor is a great idea. Also increased water changes are what's needed to help his fins from getting infected.
Your decor idea is a great route to try. They like caves, floating plants, betta logs (all my bettas LOVE their logs). The most favorite thing for my males is a single Indian Almond Leaf (IAL). I float one on the top of the water in any various corner of the tank and they immediately use them to blow bubble nests. I know IAL isn't easy to come by, but I'd venture to think that anything to provide a floating cover would work. They love floating plants because Bettas love the surface. All my males and females use the floating plants to rest in. Especially my male crowntail. He gets tired because his tail is so big. The first time I saw him basking on my floating wisteria I thought he was dead! He scared me! But he was just happy and relaxing at the top. I'm sure a little more interesting things to do and work on will help alliviate any boredom and tail biting.