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.
Nice idea Robin. How about adding this? It may save a little directory locating and typing time...
filename="$(CocoaDialog filesave --title 'Save a copy' --with- directory `dirname "$TM_FILEPATH"` --with-file "$TM_FILENAME" )" [ -n "$filename" ] && cat >"$filename" rescan_project