Are the pictures you posted earlier on the most recent ones of him? Do you have pictures of him before the infection?The black tips are part of his natural color! His fins used to be double the size and a rich black on the edges! But yes, so far no regrowth or sign of infection decreasing. I am debating whether or not to continue for a little longer or just try the medicine I have on hand
I'm baffled that you've been dosing him with salt for two weeks and are not seeing positive results yet.
Have you been doing daily 75% water changes with vacuuming included? Cleaning his filter in the bucket of tank water and wiping the inside of the glass in the tank? Also, replacing the salt you've taken out with fresh salt?
I had a fin rot problem on one of my bettas recently, for a week and a half I added 2 TBSP on salt (he's in a 5G) and did daily water changes, removing 75% of the water and gravel vacuuming. Cleaning the filter and glass in the process. I temperature matched the fresh water and added the replacement measurement of salt to the bucket, after mixing it up in a cup of water with a spoon. I cut his feeding down to 3 pellets a day and kept his tank at 80F. He had no signs of active rot on his fins by this point (@JuiceBox52 clarified it for me) and I noticed tiny, transparent growth at the end of his fins. Now I'm doing 20% water changes and gravel vacuuming to remove the salt from his aquarium. Still wiping down the inside of the glass and rinsing his filter and normalised his eating routine, varying foods etc. He's doing really well. Fins are showing healthy growth and i don't feel so guilty when looking at him.
I'd stick to salt personally, unless someone more experienced with this issue says otherwise and advises you to use an antibiotic. Just persist with those water changes, correct cleaning routine and salt measurements.
Could really do with the more experienced members on this forum to give their opinion for you going forward.
Hang on in there!