Unscrewing a rounded allen bolt
I'd try a screw remover like these. You'll need to either get a set or bring the bolt in to make sure you get the right size (Sears sells their own version that's supposed to be excellent). The good ones have a left-hand thread and a small cutting head to bite into the bead of the bolt; this looks like a really good candidate for a screw remover since the head is nice and large and has enough material for the remover to be able to grab.