NEW background check question

Situation: Landed IBD SA at a BB. Worked at another BB PWM freshman summer, did NOT put it on my resume. It was unpaid, so no W2/tax forms to verify that I worked there. Also have lost contact with the people I worked with.

Question: Should I list it on the background check employer list? I think honesty is the best policy, but I see potential problems either way. If I list them as an employer, and it doesn't show up on the background check because it was unpaid, it's a red flag. If I don't list them as an employer (which should theoretically be fine since it wasn't on my resume), and it does show up, they'll wonder why I left it off in the first place.

Before you yell "SEARCH", I've been through the 20+ (lol) threads on background checks. They're either about trying to verify an unpaid internship that someone DID put on their resume, or about leaving an unpaid internship off the background check that they got fired from or something.

Any advice? Thanks.