Putting together a high altitude/low temperature radiation meter might be difficult. Most of this is because a Geiger-Muller tube depends on the interior gas pressure to be within a certain window to work correctly. If the temperature drops too low (i.e., pressure drops too low), then they just don't work. It might be possible that someone has overcome this problem, but I'm not sure.
Other options might be to add a heater module if they have the room for that kind of extra mass. Maybe fly at a lower altitude(s) and figure radiation levels higher up using mathematical models?
This is a pretty tough problem being proposed.