Again, I'm not advocating for the tests, and I have no detailed knowledge of their statistical utility, if any. Could be just some more BS the HR ladies made up to justify their existence.
My gut feel is that the best (only) avenue to challenge the practice would be to show that it's unfair, as opposed to an ADA violation. But unfair isn't necessarily illegal, so you'd probably have to tie it to some other law, ie discrimination for something such as gender, race, age, etc.
These things have been around for a while, you'd think some industrious shyster would have already at least considered a class action by now if there was any merit to it.
Also... you might have trouble getting pilots to join you. If they've reached their career-destination job, they probably don't care anymore. If they haven't, most probably don't want to volunteer to be on a very public list of people who fail personality tests and also sue airlines. That list would end up taped to the wall in every airline recruiter and HR manager's office. Litigious people are not a protected class, you can discriminate in hiring against them as far as I know.