The following link to Neil Ashby's 2003 paper on Relativity in GPS:
http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf
Of note: "Also, experimental tests of relativity can be performed with GPS, although generally speaking these are not at a level of precision any better than previously existing tests." - page 6
"At the time of launch of the NTS-2 satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was recognized that orbiting clocks would require a relativistic correction, but there was uncertainty as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic effects were truths that would need to be incorporated [5]! A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation.
Also of note is the Hafele–Keating experiment which as far as I can tell is only different from what Le Repteux was originally describbing by the clock being onboard a satellite vs an airplane.
Suffices to say deriving time corrections to account for relavistic effects is clearly possible and one of the ways relativity has been validated.