Good thinking, @jpmeijers, I didn't even consider the known locations of gateways (added now), but was only thinking about the known location of the (mobile) application showing the position.
I was wrong about the sign. When using decimal degrees it is needed, for use around the equator and the prime meridian (the "zero" values for latitude and longitude). The 50 km I mentioned was simply the 111 km "weight" of 10 times 11.1 km for the first decimal, divided by two to ensure one can figure out the sign. But that won't work. And maybe one also needs to divide by two as the range is a radius.
Also, I just read that most GPS devices will output strings defined by the National Marine Electronics Association protocol (NMEA), which gives one latitude and longitude in degrees, minutes and decimal minutes.
So: maybe one can still come up with some 2 byte value that can be enhanced by the application. But it's not as simple as just sending the 4 decimals of some decimal degrees value, and the range will be smaller than the 50 km I first assumed.
(To be continued.)
