American says a maximum of two batteries are allowed on board per passenger. So, three DJI
Mavic 3 Pro batteries will not be a go unless your wife puts one in her carry-on bag. Any additional battery bank also would also have to be considered. The one you mentioned falls under the maximum capacity, so it could be carried as her second allowable battery.
The phrase "with airline approval" is a bit bothersome. You might want to make a written or emailed request for approval for the specific items you want to take and take that with you to show the gate agent of cabin crew if they raise an issue.
I remember having batteries detected and checked for a flight in Italy 20 years or so ago. I've heard nothing about battery issues in carry-on baggage lately.