A depleted battery can make a fire the same as a fully charged battery. If a fire starts on the plane, it's a fire, and they have ways to address carry on's that catch fire (happens very rarely). It's the integrity of the battery that really matters. Hence your should be aware that batteries will show signs of poor integrity over time, with the sides bulging out, swelling up. That's what matters, battery integrity, IMO.
I always fly on airlines with my batteries fully charged because I will often go direct from the airport to a flight location to fly. TSA has never asked any questions about charge level, in over 100+ flights in the last 4yrs. The charge level isn't a concern.
If the OP is doing two trips, I highly suggest he invest in a good backpack for the P3P and carry it on the airplane. Forget the hard case! I only check things that are replaceable when I arrive. Everything else I'll carry on. Hard cases are heavy and awkward while on vacation (unless you've got one of the DJI light weight hard shell backpacks). You cannot hike or bike with a typical hard case, which limits your options to fly in the best locations. When someone sees you carrying a hard case in the middle of the mountains, people get suspicious. However a backpack is not. The Think Tank Helipak is the best one IMO, if you want something with quality. Otherwise, you can buy cheap drone backpacks for a little as $35 from China that work fine for about a year or so.