I have a betta that I have experienced similar problems with. I haven't actually ever seen parts of his tail but just noticed each evening that more was missing. All I had in the tank was a whisper 10 filter which has no sharp edges either. I tried melafix for almost 2 weeks, added a heater and an air stone but nothing helped. I finally decided that it wasn't fin rot and moved him into a breeder box inside my 29 gallon. His tail is now healing without medication. He always liked to swim down under the filter and then back up behind it. I guess he was tearing his tail and fin on it some how. I want to put him back in his tank but can't figure out what to do to make it safe.