It only occurred to me way, way too late that another way to approach this project is with a Raspberry Pi Zero W. The add-on board would simply provide 5 volt power from the 15 volt primary supply and use 5 GPIO pins to drive the MOSFETs in the same way as the existing project.

The advantage to this would be that the lack of a user interface is less of a handicap (since you could ssh in), you could use WiFi and NTP instead of GPS, and the whole thing could be done with Python scripts.

Cost-wise it would be a wash, since the GPS hardware is about the same price as the Pi Zero W.