While the PSU is definitely small and cheap, if it were bottom-of-the-barrel junk, Gateway would probably not be as large and successful as they are. I would not run a HD7770 on it, which I would probably be willing to do with a Seasonic or FSP 300W PSU; that said I think you will be okay with a HD7750.
Using the PSU calculator at http://www.extreme.outervision.com/PSUEngine
I threw in an extra fan, card reader, 4x4GB of RAM, counting your drive as a non-"green" drive, added 10% capacitor aging and came up with 275W. If it's 2x8GB sticks, and a "green" drive, it drops down to 263W. In each case, the minimum was 50W less. I think you'll be ok.