I want to understand something here...it sounds like all your cats are living downstairs, is that correct? I understand keeping them seperate, but I think your stray doesn't like being confined to one part of the basement. Seems that way with most strays in threads I've been reading, for the most part. Is there any reason why your cats can't come upstairs? I think most kitties do not like being shut off from any part of a house, including my own 2. They must have access to my closet, for example, so I've given up keeping my bedroom closet closed, and same with my bedroom & bathroom doors (just keep the toilet lid covered). You may also just want to swap the cats around and "rotate" them to different parts of your home at different times so it's equal access to all.
If your stray likes interactive toys, your kitty may just need to burn off some excess energy after being in an enclosed room for so long. I know I wouldn't like to be couped up by myself if all I've ever known was being outside, free to go wherever I wanted.
I don't have experience with strays, though, so hopefully someone with more experience can help you through this better!