Yes I'd leave him in there and treat the whole tank.
Tail rot is usually a water quality issue so I'd up your frequency of water changes and monitor the parameters daily to see what's going on. I can see from your signature that your 40 gallon has the cory in there, and you have excellent filtration, but an ammonia spike can sneak up on you, maybe a piece of uneaten food that you didn't see, or a over-vigorous vacuum of the substrate stirring up a lot of mulm, poo, and decaying matter.
Whatever the reason, pristine water will go a long way to help your cory get over this.
When I had an Oto that was injured by a Macrobrachium sp. shrimp I did daily 50% water changes every day for two weeks and dosed the tank with a capful of Melafix after each change.
The Oto had significant injuries; his caudal(tail) fin was completely nipped off close to the body, there was a big chunk out of the dorsal fin, and his nose and face were torn up and bloody.
With those huge water changes every day he was completely healed in two weeks. I know the pristine water was vital, but I also do believe the Melafix helped him avoid an infection after this very traumatic event, as it has mildly antibacterial properties.
I don't use Melafix alone if I know I have a serious infection in a fish, but it does help with healing and I firmly believe it helps prevent infections in injuries as fish are healing. I believe it will help with the tail rot in your cory.
I know from experience that it kills bacteria because I actually started having ammonia traces in the tank: my beneficial bacteria took a hit and wasn't able to keep up with the bio-load. This was after the first week of doing daily 50% water changes and a capful of Melafix every day in my 40 gallon tank.
I hadn't added to the bio-load, or neglected uneaten food, or any of the other reasons that would explain the appearance of ammonia in this tank that had been cycled for almost a year.
I kept up the water changes and Melafix for two weeks and then in the third week I backed off to every other day on the 50% water changes. I could see so much improvement and no signs of infection so I stopped using the Melafix. My beneficial bacteria recovered in about four or five days so it hadn't totally nuked all of it.
I hope your cory feels better soon. Post pics, if you can, so we can see the extent of the tail rot and chronicle his recovery. Keep us posted.