To my knowledge, there is no background check associated with an ATP checkride.

I also know that there is no recurrent background check for people who already hold an ATP.

Where they get you is on the medical renewal. They ask about criminal convictions and arrests, and they will deny a first class medical if you have a significant conviction.

I would assume that the FAA would not dare to withhold a medical (or ATP cert) on moral character grounds without a conviction for something serious.

For example, these sort of things would probably not be an issue with the FAA

- Cheating on your spouse
- Hanging out with known criminals
- Radical political views
- Failure to attend church, or attending the wrong church

But watch out for the TSA, they can pull your tickets for national security reasons...which means whatever they want it to mean.