The only way this will occur will be through lobbying.
A Public PR campaign will never work.
The public doesn't care and we know exactly why by example.
Not too long ago(2011 if I remember correctly), the airline taxes were non-existent when the FAA re authorization lapsed. The airlines raised their fares immediately to make up for the loss of tax revenue.
So basically, this argument doesn't affect fares one bit. The only effect with the taxes is who gets the money. The airlines don't have the ability to raise fares at will, but the cost of the ticket won't change. The only change will be how much the government gets versus the airlines.
At the end of this year, when they see the record profits at DL and UA, they will likely just raise taxes and take away the profits in the future.