An error code for a mainboard doesn't help without telling us what board it is. And the manual should list what the error codes indicate.
A battery backup shouldn't cause any significant EMI. The battery and related circuitry isn't even active unless there's a power outage. There was one model of sort of UPS unit that fit INTO the case a while back. My UPS is only a couple of feet away from my computer.
You're saying it only bluescreens when it's put into the desk, if it's outside the desk it works fine, and you can consistently repeat this behavior?
Are you bending cables at all when you put it into the desk? It's conceivable if you were pushing down hard on like the video cable, it could be flexing the video card in the slot for instance.