When I bring home new bettas that may have fin rot but I am not sure, I use a half dose of aquarium salt in their tanks and observe them for a while. If it seems that they are getting worse I use JFC which has worked wonders for me. I'm glad to see that more people are using it now. It took me a while to stumble upon it in the pet store because most places recommend different meds but this is the only one that seems to work for me - I love that stuff!
BTW, it is very possible that he could have bitten his fins- many will when under stress, which is why I would just use a little salt as a preventative and wait and see. Good luck with him! He's very adorable :)