Yes, it's absolutely possible a "known good" CPU will work in one board and not the other. It is a 'microcode' issue, but it's not user serviceable. It's generally a fuse(d) bit in the factory. While the functionality to access and possibly (unlikely) burn these fuse(d) bits may be available, no...
Bummer, it just sounds very unusual as the graphics is pretty tightly integrated with the other chipset functions (it's on the same silicon AFAIK). Not intimately familiar with the 945 series, but IIRC the northbridge contains graphics.