I share your skepticism about the GPS/frame of reference explanation for the FTL neutrino finding--not that I know anything about it.
But I did see a summary of the original paper reporting the apparent FTL finding. It tells me that the two timing units were synchronized and calibrated at the Swiss Meterology Institute, then moved to CERN and LNGS. A third device was then set and moved back and forth to assure no discrepancy had developed.
So the orbiting satellites relative to the departure and arrival locations appear to be irrelevant, so far as I can see. The timing was done at the sites with exquisitely calibrated devices, not with live GPS signals.
"If you can keep your head while those around you are losing theirs, you can collect a lot of heads."