There are to reasons airports need to charge rental companies if we ignore 'they are abusing their local monopoly'.

The first is congestion due to the pick-up process. Clearly, this affects both the new players and the old players. The second reason is land-use by the parking lots in which available cars are stored. This doesn't affect the new players. Now, airports might be charging for both, and simply be afraid that the cash-cow of charging reasonably for the land-use of parking lots is going to disappear.

For a real answer, you'd need to actually compare the difference in cost between congestion and land-use.