I took apart the GPS. The unit passes power to the camera through some ferrite beads. The GPS is an off the self unit that is just wired to the custom power circuit board. It runs on 3.3V. USB power is 5V, so it does not connect directly to the USB power. It gets its power from the camera.
The only thing I could see that might have been the issue is that the contact for the USB power between the GPS and the camera did not stick up as much as the other contacts. I bent that pin up a bit to match the others.
Since I did not see anything I could definitively point to as being the problem in the GPS, I opened up the camera. The GPS connector on the camera is connected to the rest of the camera via a flex cable with ZIF connectors at each end. I did not see anything wrong here either but just in case there was a contact issue with the flex connectors I reversed the flex cable so that each signal would be on different wires in the flex cable, and the connectors would contact at different points.
I now have the camera powered up through the gps unit, and there does not currently appear to be a problem with it.
Now I have the nasty task of cleaning the old adhesive off my windshield so I can re-install the camera.
Before I do though, I will put the latest firmware on the camera.