A quick web search lists 20-25 W for the Radeon 7000. A Geforce MX draws 10 - 15 W max. depending on the design, maybe that one'll work.
I'm not sure if power consumption of the video card is the real problem here, as the cards only draw maximum power when operating in 3D mode, which is not the case when booting the machine. Ususally power problems with video cards reveal themselves only when playing demanding 3D games.
You could hook up another power supply for testing, but there's a catch: Brand name machines often have non-standard pinouts on their power supplies, even when the plug looks like a standard one. So do not connect a PSU unless you're absolutely sure the pinout is correct!