This sounds tricky! 
So PX4 does not do (or require) calibration before each flight meaning you should be able to calibrate it beforehand.
However, the estimator (e.g. EKF2) on PX4 requires some time to initialize, as well as the GPS needs some time to get a 3D lock. Therefore it would take too long to boot PX4 mid-air. Instead, you should be able to initialize everything on ground and then arm PX4 once the arms are unfolded, etc.
This might still be tricky because the estimator might not be tested/tuned for high accelerations that you might encounter. Also, the tuning on 5000 ft might be slightly off due to the thinner air.