This is not quite correct. Basic (to those who know it, but not basic to those who have never considered it) tax incidence theory tells us that part of the tax is paid by consumers in the form of higher prices and part is paid by the seller in terms of lower revenues received. This is true of whether AA collects the tax and remits or or if you were to pay the taxing authority directly. In other words, AA has an incentive to fight to tax because it results in lower revenue for them--because they cannot pass on the entire tax to the traveler.

Now there is one situation when AA can, in fact, pass on the entire tax. This is when the consumer is completely price inelastic, meaning they cannot change their consumption, no matter what the price change is. This, unfortunately, approximates the case of those who purchased their tickets already, and now must pay the ENTIRE tax increase. AA can pass this on because they already bought the ticket. For future sales, however, AA will have to absorb some of the cost of the tax, and, therefore, should (and did) fight the tax increase.

This wasn't very clear as it usually is an entire chapter in a public economics textbook, but hey, no one wants to read that.