Do festivals screen your footage to see if the music matches with any copyright protected material?

Depends on the festival. Usually they leave it up to you. If you get sued, that's your problem.

who's gonna be able to tell.

The copyright owner.

No human ... would be able to recognize the source of a classical piece.

That's why they have computers.

The songs I'm thinking about have been performed thousands of times all over the world.

The classical "piece" may be old enough to fall under fair use...however the PERFORMER(s) still have a claim on the recorded performance which can be protected by ASCAP or BMI or whoever holds the copyright.

Is this slight ethical discrepancy really anything to worry about?

It's stealing. Period. You may or may not get caught. The copyright holder may or may not care. Bottom line is, they have every right to sue you if they find out.

If you frequent this sub, you may have seen all the Video Essayists getting shut down and sued in the past few weeks. There's the girl who got sued over her makeup blog for playing 5 seconds of electronic music.

The shit happens. Want to risk it? That's up to you.