Betta Fish: Please help me save this betta!

For some background. I have had many tanks and have experience with thriving planted tanks, but I’ve never had a betta fish nor had the desire to get one because I’ve never found a pet store that actually cares about them. Recently I went to petsmart and a betta fish was by itself in a corner with a sign that read “I’m sick, I’m being cured and will be ready to adopt soon.” My heart broke for this fish, I’ve never seen such a rough betta… all of his fins are almost gone from what I’m assuming is ammonia (I tested his cup, it was at 8.0 ppm) and he was laying on the bottom barely breathing. I talked to the store keeper and she said they actually do nothing and they just die, the sign is basically there so people don’t get mad (truly terrible). So she let me have him. Now I am semi-prepared since I’ve had tanks, but unfortunately we just moved homes and my hospital tank bit the dust. So I have a new 5 gal with one plant and catappa leaves and a heater, and the sponge filter had active bacteria or whatever in it from my other healthy tanks to attempt to boost this in fish cycling. Regardless it’s better than what he was in since his cup was so toxic.

But I need help! Any advice on what to do???
There is a picture of the 5 gallon tank, it has shallow water so it can swim up (it barely can). Is this shallow enough? The tank is 80 degrees, has 1 catappa leaves and a plant for him to hide. It has the sponge filter but it’s not on because the tank is too shallow, but an air stone is on. Does it need a filter if I’m already going to be doing heavy water changes since it’s basically cycling with fish in? I put the air stone inside the sponge filter. All params at 0 since it’s new tank, pH slowly lowering down to 7 (I live in FL so the water here is very hard).

I’m not sure what’s wrong with him either besides barely any fins, any advice? I’ve attached pics of him but it won’t let me attach a video. He mostly lays on the bottom of the tank, occasionally will swim around to try and get to surface but realllly struggles to stay up right and swim. He did eat a pellet (not sure if I should feed but idk when his last meal was and he kept trying to eat things). Don’t want to put any meds in until I have more of an idea bc he’s already so stressed.

Thanks everyone!