Since Perl is not (usualy) multi-threading (and when it is Tk is not thread safe), sleeping will delay TK's processing of events, meaning that you still can't see that there is a double click from inside a sub called from the single click event.
Now that I think of it though, $mw->update() should let Tk notice the double click, but I don't know how you would catch that from inside the sub, and I would think it would beg for odd timing issues to crop up.