If Apple did write the firmware that the FBI wants and then signed it, would changing the device UUID hard coded into the firmware not invalidate the signature?

If Apple did write the firmware that the FBI wants and then signed it, would changing the device UUID hard coded into the firmware not invalidate the signature? Is the concern that there are somehow other signing keys in the chain of trust that exist outside of Apple that would make it a general exploit or is the concern that they would be a much lower threshold for getting this sort of thing? Apple might have a point if it is the latter, but if it is the former, the security of the iPhone is already compromised.