Thanks, Menagerie, I was going to ask those same questions!
A big factor for me, when one of my previous bettas had fin rot, was temperature fluctuations. Bowl temps often fluctuate more than a tank with a heater.
Only once did I see a black piece of fin when my betta had finrot. Usually it's just frayed-looking. When it starts to heal, the edge often turns whitish. My betta tore himself up a little, playing in the sponge filter bubbles. I didn't add any meds, because it wasn't really fin rot, just torn, and he's healing well now.
Without knowing details of what happened, or what your parameters are, I would say just watch him, and do a few extra water changes to keep the water clean so infection won't set in.
sugarloves, your betta is gorgeous! He could almost be a twin of my betta