Given your price range, I'd say your best option is to buy a used ultralight computer and remove the hard drive and battery. That should get you under three pounds.
You sure you don't mean 3 kilos? (~6.6 pounds?) That gives you a lot of leeway. You can even have a hard drive, processor, battery, and probably a discrete graphics solution in a computer that weighs that much!
Seriously, the only notebooks I know of under 3 pounds are really pricey. (Panasonic Toughbook eLite)