Damages should be fairly easy to prove. Hire a couple of experts, track profits from similarly situated stores, and calculate out the lost sales.
For example, they have one in DFW Terminal C. Adjust for passenger volume and employee levels for that location. Confirm with other locations, fine tune the formulas, and you've got your evidence. The length of the contract is whatever it is, so multiply that out. The airport probably has some optimistic traffic growth projections they use for their bond offerings; use those against them as well.
Proving the amount of damages isn’t the issue. Governmental immunity is the issue.
As in every other state, the state and local governments have sovereign or governmental immunity from suit unless that immunity is waived by the state legislature. The legislative waiver of sovereign immunity in Texas is relatively limited when compared with many other states. There is no waiver of sovereign immunity in Texas to allow a private entity to sue a city or other local government for damages for tortious interference or other commercial torts.
That said, it is possible Chik-fil-A could have a federal claim against the City for violating its civil rights under section 1983. I can’t say I’ve ever seen a successful tortious interference 1983 claim, but there’s a first time for everything. It would be an interesting argument.