You need something super light, I don't think you've got much chance of a zoom at that price point, but you can always crop in post production. Check these out: http://dashcamtalk.com
I'd wager a möbius or innovv would do the trick for sensible money. I mounted my point and shoot, with terrible shake results, but the tiny light innovv camera, although cheaper and less capable, produced way better results. You can also eliminate some shake in processing after, but it can't perform miracles.