If a person requests their criminal background via fingerprints from the FBI

Typically not. I have been requesting a criminal record check from the FBI for many years now and never had something like that happen. Wonder if any of the other attorneys had a different experience.

FBI doesn't like to have much to do with Immigration agencies and will not share any information unless a matter of national security or an "aggravated" felon (think drug dealer, smuggler, Russian/ Ukrainian crime syndicate - you get the picture.) FBI criminal background report is the most reliable way to determine whether someone has been deported in the distant past (much more accurate than FOIA request results from DHS).

Very little chance.