It sounds as though what you really want is a "Save a copy…" command, which
makes a new file containing the contents of your current document, rather
than renaming the current document.
TextMate doesn't have such a command built-in, but it's easy to write one.
Here it is, attached. I've bound it to apple-D, which doesn't seem to clash
with anything, but you can easily change that in the bundle editor.