The Federal Aviation Administration's mission is to watch closely for maintenance lapses and other safety problems at U.S. airlines. But last summer, two FAA inspectors allege, Southwest Airlines was trying to pick and choose which inspectors would do the watching.

Annoyed by one FAA inspector it considered adversarial, Southwest complained in June to local FAA managers, sharing the airline's detailed records about the activities of that inspector. That incident and others like it, described by the two inspectors, have now...