My charge controller boards have arrived and they actually work - despite my soldering skills! Photos are attached.
I'll probably design a second version with a few minor modifications - mostly a smaller Schottky diode, bigger pads for that pesky Inductor (the L1 silver component near the top left), and better labelling.
This charges the supercapacitor to 5.2 volts from only a couple small .5v solar cells. I now need to determine how many solar cells I need to provide sufficient current for the 2 minute transmission. My current draw is 3-5mA when idle, 35mA during GPS acquisition, and 42 mA when transmitting.
I hacked around the issue of the transmitter interfering with the GPS. I added an NPN transistor to act as a switch to use the Arduino to turn the power on to the transmitter only when it's needed. Unfortunately, when I turned it on, the interference kicked the GPS out of sleep mode (jumping it from 5mA to 30mA, or 70+mA for everything). So I added a second switch to turn the GPS off when it's not needed. This adds a bit of weight and requires a longer cold boot GPS fix every 10 minutes, but that shouldn't be an issue at 40,000 feet.
If I can get the antenna set up and everything tied together, I might see if the USU folks will let me fly one on one of their balloons this Saturday.