I spoke about this with our software team this morning, and they suggested you try measuring the voltage of the capacitor, as it may be dropping below the Ping360’s 11V minimum requirement.
If the voltage is fine and it is in fact just an issue with the startup then it may be worth looking into getting a relay which controls the Ping360 power, which you can control via the Pixhawk. That way you could power on the vehicle, then turn on Ping360 power once the input power has stabilised. With the current stable Companion software you’d need to reboot the Companion computer after the Ping360 is turned on in order to detect it, but if you’re using the Companion Beta then it should be detected automatically (it searches for Ping devices regularly, instead of just once on startup).