I thought about it some more, you might be able to do this entirely with an off the shell SDR.
If you have a device that can record multiple bands simultaneously, and preserve timing between them, then you may be able to record both your target band and the GPS band and then use software later to reconstruct the time out of the GPS recording.
This is a very interesting idea. I think I could use a BladeRF, it has two input channels. I hope the two channels are perfectly synchronized, by glancing at people's projects it seems that they are.
If you have a device that can record multiple bands simultaneously, and preserve timing between them, then you may be able to record both your target band and the GPS band and then use software later to reconstruct the time out of the GPS recording.
See: https://www.rtl-sdr.com/rtl-sdr-tutorial-gps-decoding-plotti...