The recommended water change schedule (on this forum) for an unfiltered 2.5 gallon is one 100% and one 50% water change per week. So you may want to increase your water changes, especially since the little guy can't shake the fin rot.
If clean, warm water isn't sufficient to cure the rot, generally, the next step is to use aquarium salt. However, bettas are sensitive to salt. Using too much, or for too long, can cause kidney damage. Also, I worry about using it if he is a baby who hasn't matured yet. (I don't know if he's truly a baby, or you just refer to him that way!
How long did you use the aquarium salt, and at what dosage? It has antimicrobial properties, and may help cure the rot - which is often due to gram negative bacteria......but you don't want to use it long term, or in high dosages. If it isn't effective, though, he may need a gram negative antibiotic, but let's see if there's another alternative first.
In the meantime, I recommend upping the water changes. When my guy had persistent rot, I did 100% changes every 3.5 days (twice a week).