How often do you perform a water change? I hadn't until now...the petstore lady said to just change the filter unless the water got dirty or stinky & it was still crystal clear.
What percentage of the water do you change when you perform a water change? n/a
What type of additives do you add to the water when you perform a water change? I have to add water due to evaporation & then I add a betta water conditioner with the new water.
If you haven't changed the water in ...five months
... (pet store people are notoriously stupid when it comes to bettas..) there is going to be a massive buildup of toxins in the water - not only ammonia but all kinds of nasty. What happens then is that there's a kind of chemical process in all that concentrated poison that turns ammonia into ammonium and stops the fish from dying quickly. However, all the other gunk in the water -will- slowly weaken and eventually kill your fish.
But if you change the water suddenly, two things can happen - 1, your fish goes into shock at the new water chemistry especially if you don't acclimate him slowly. 2, tons of built up ammonium (not harmful) reverts right back to ammonia (harmful).
I don't think this fish is marbling after 5 months.. I could be wrong about that, however... and his fins look awful. Those holes are caused by ammonia, and possibly bacteria.
I'm glad he's not in his home tank -- glad you're cleaning it out. Do the first few water changes in the small bowl very slowly - 25 % at a time. Do one of those every 15 mins for an hour (or slower/smaller amounts over a longer time, if you can), then keep the changes regular from that point.
In a gallon bowl, you'll need to change the water 50% at least twice a week, as well as 100% once a week. That's the problem with little bowls - poisons build up quickly. In fact, to help his fins heal, he'll need really pristine water (100% every 2nd day is what I do, with advanced fin rot in my 1.5 g hosp tank, you may prefer to do daily in a 1 gallon) and some salt.
Get some rock salt, 1 teaspoon per gallon and predissolve this in conditioned water (Prime's the best water conditioner there is, imo, and great value for money) for every water change.
No more than 10 days of that. See if his fins improve. Keep him in the small tank for all this, it'll be easier to clean regularly.
I wouldn't use any of the bettafix, melafix stuff on any of my fish, ever - there's a risk of injury to breathing organs and it's just an antiseptic with limited uses anyway. If there's a serious infection, I use salt early on, or a proper antibiotic if it's fast or advanced.
If you can get some ketapang leaves (Indian Almond Leaves or IAL) and darken his water with one, this will provide a natural betta-friendly antifungal and antibacterial quality to the water. They're pretty cheap for a large bag, usually, and can be bought online.
Is that white colouration fuzzy/soft looking at all? Or is it like the rest of his scales, only pale? That will help figure out what's happening with it.