I don't agree
When a satellit is inserted in orbit you can easy measure the time dilation on board, you can also easy set up software to syncronize the dilation.
So fare you don't need to know very much more, - also not the math behind relativity, but only how to program a computer software, able to eliminate the time dilation orbit by orbit, and the GPS system is working fine..
You can also easy explain the cause of the dilation by combining SR and GR contributions, but this is not very important..
Therefore if there is a period with missing SR signal each orbit (which you are not aware of) , (when moving north) you can easy replace the missing SR signal with a estimated GR signal, which make you believe the orbit is different than it really is.