Quick update - I added a 1A fuse between the battery and the USB charging and protection board, and that is how I plan to leave it. I took these measurements after installing the fuse to see what voltages the charging protection board would cut off the battery:
Battery Low Cutoff 3.04v (with 1A fuse in series with battery)
Battery After Charge 4.22v (with 1A fuse in series with battery)
These work for me, but I am still not really happy with the performance. I tried a 5V regulator board and a 5V fan but the performance was only modestly improved, if that. I think I need to go bigger on the fan to have adequate solder fume extraction, but this will work in a pinch.