Round trip Seattle-Maui in December with P3P and 3 batteries in DJI hardshell backpack inside a 22" Costco rolling suitcase just for convenience, as we had our laptops and personal stuff in our normal backpacks.
I was pleasantly surprised that the X-ray techs knew exactly what it was on their screens, and didn't do any add'l screening at all either way. One just said 'nice drone!'
The two-battery limit is for those over 100 Watt Hours, and the P3's Batteries are 68 Watt Hours (44 for the RC), so there shouldn't be a limit on the number of P3 batteries you can carry. Not sure about the larger P4 batteries?
Pack Safe – Lithium ion and lithium metal batteries, spare (uninstalled)
Professional - Batteries Watt Hours
It's almost certainly wise to print out the FAA sheet above, and a spec sheet for your batteries, in case you do end up having a disagreement with TSA or the airline.
It's a very easy thing to do, and given that all it takes is one guy questioning your watt-hour situation to ruin your day, having those tucked in your bag could hopefully help fix it quickly if it ever comes up.