Hey! Good on you for what you did, stepping up for this animal and showing compassion for it when no one else was. (I don't remember if it was mentioned, does the betta have a name?)
This actually really reminds me of how I got one of my bettas, Rigsby, and I can tell you a bit about how I got him and what I did to help him on the road to recovery.
Rigbsy was the fish of the people who live in the room next to mine in the dorms. The first time I saw him, several months into the semester, I was horrified. I could describe it, but:
Lettuce floating in the water, a layer of gunk all across the surface of the water, a smell that literally brought me this close
to vomiting... ugh. To make a long story short, I basically did what you did: took him, set him up in a spare tank I had at home, and started nursing him back to health. Eventually, my neighbor passed his care permanently on to me, I think after seeing just how much he had changed from the fish that she knew.
Anyways. When I first brought Rigsby to my room I had no idea whether he would live or die. Fin rot, swim bladder disease, extremely faded colors... he was in a bad way, not surprising considering what he had been living in for several months. I immediately started doing 100 percent water changes daily, adding in a liberal dose of Stress Coat with each one. One thing I really believe helped, was getting him in a real tank, with plants and room for him to swim around, something he had never been able to do before; he absolutely adores his hammock. I know you said you are thinking of getting him a new tank and I think that would really help! Sitting side by side with my other fish also brought some spark to his swim; he and Odysseus are buddies and I really do think the stimulation they provide each other helps keep them healthy.
Betta fish are hardy and can survive so much. He already looks so much better and I believe he will really flourish under your care! I know it might sound corny or cheesy, but I think more than anything Rigsby was able to tell that I cared for him and that seeing me meant he was safe and secure and was going to get what it was that he needed. I think it was this knowledge more than anything that helped him to heal. I think for now I would pass things off as "looking after him", as you said, which is what I did with Rigsby until he was officially given to me. With Rigsby I really emphasized the fact that he was sick and that I knew what to do to help him get better. If your roommate says nothing, especially when the time for you to move is approaching, I would just assume he is yours. Your roommate may come around and like not having the responsibility anymore, like Rigsby's did, as awful as that sounds.
Sorry this got a bit long! I hope it wasn't too aggravating or dreary to read. This just reminds me so much of Rigsby and it's close to my heart. I am so thankful you stepped up for this fish and I really admire what you've done.
This is Rigsby now, and you wouldn't know he's the same fish, in looks or behavior. I think your guy can live long and prosper in just the same way! You're doing a great job, and you're definitely doing right by the little (or big!) guy. Let me know if you have any questions and definitely keep us updated! :)