I modified the Arduino program to output the raw time for both of its interrupt pins. In lieu of a working gamma detector, (as the op amps haven't arrived yet,) I used a button press to simulate a pulse.
Next, I need to have the Arduino read the serial data from the GPS, which includes not only location but also time of the last sent pulse. Considering how easy it is to implement timing measurement, I think I'll try to use one microprocessor rather than two. If I can get everything, including networking, working on a single Arduino, that would reduce a lot of the cost I originally estimated.