According to Allan Odgaard:
Ups... I had the run loop wait for the distant past, effectively creating a busy-loop. Here's an updated version: http://macromates.com/textmate/files/TextMate.service.zip
You said it only supports one edit task at the same time. Does this mean that if you already have TM open (which I have all the time), the menu doesn't work?
I confess I have started it manually, not wanting to lose all my session by logout/login, it may make a difference.