You're a sudoer (i.e. admin) on the server? It's not possible to use sudo in BBEdit AFAIK. (You actually download as the user logged in, and you're re-uploading the file after making changes, again as that user.) You'd need to use a local (local to the server, that is) editor then in a Terminal ssh-session. pico would probably be installed. If it's only about small changes, that'd work just fine. (Actually, pico is quite nice and does a lot of things well.)
You could, of course, use an ssh-shell to temporarily change the file's ownership and permissions, edit with BBEdit over SFTP and then, when changes are saved, revert the ownership and permissions.
It all depends on how many changes you want to make to how many files, really. If it's only about small things and a couple files, I'd go the pico-route.