Regardless of filtration or not- water changes still need to be made on a regular basis-
The behavior you are seeing is most likely related to water quality issues and build up of DOC's and ammonia
What is the water temp? forgot to ask....
In a 2.5gal filtered tank I would recommend twice weekly 50% water changes-1-water only and 1-to include the substrate by vacuuming or stir and dip
I would start by making some 50% daily water changes for 3-4 days, give your filter a rinse/swish in old tank water 1-2 times a month and when the flow slows
To help in healing and re-growth of the fins as well as prevention of secondary infection from the open wounds I would stay on top of the water changes.
Depending on how bad his fins are you may need to QT and do some 100% daily water change along with aquarium salt 1tsp/gal for 10 days....can you post a pic?