I think that may be the Stelvio…
But to answer your question, it sounds as if any time on the bike would be useful. How come you can't bring yourself to get out on your geared bike?!
Hill training on the single-speed would be OK, but probably not the most targeted and effective training. Assuming you've got a do-it-all gear on it, then you'd be riding up the hills over-geared, which would help most at building strength.
On a climb like the Tourmalet you'll have to be putting in an effort sustained over 1.5 hours or whatnot. So in my book it sounds like it'd be a good idea to put in some time building general endurance and cardio-vascular fitness too. Long rides and sustained climbs at tempo, not just explosive efforts.
That's a completely unscientific opinion BTW… :-)