Waco, we've done polling and focus groups on this in the past. The result is always the same: if you do what you're talking about, the public thinks "there go those lazy, overpaid pilots again." The fact is, despite the horrible situation that the AMR pilots are in, they make a ton more money than the average American, and they still will after the bankruptcy is over. People don't tend to sympathize. Safety is about the only thing that the public will pay attention to, which is why we finally got somewhere with pilot licensing requirements and flight time/duty time, but when it comes to compensation, work rules, and benefits, the public doesn't care one bit.
But, let's say your idea could work, and that people would care. Let's say the APA went on a public marketing campaign and made the public mad at AMR, so they started booking elsewhere. What does this accomplish? Bookings go down, and AMR's revenue drops. The company loses even more money. The company cries poor to the bankruptcy judge, and even worse conditions are imposed on the pilots. Sorry, but despite what the anti-union people think, labor doesn't really like to hurt our employers. It only hurts us in the end. Management leaves, because they're nothing but a bunch of opportunistic carpet-baggers, but we're still here, left with the aftermath. The strategy just doesn't work. It's counter-productive, even if you could get the public to care, which is unlikely.