My experience of the background check was that it is
extremely cursory. I was two cubicles over when my company's CFO took the call from the background-checking firm, and the call lasted maybe 2 minutes. They're really running down the right-most column of your resume to make sure the dates line up. I'm not sure if the academic side of the background check is more thorough, but I'd be surprised if it were. Neither of the people on the two ends of that phone call really has much interest in going too deep: the checker has a stack of names to get through, and the person taking the call has a job to do that's being interrupted.
One thing you could for sure do is simply call up your undergrad institution and ask if they disclose that kind of thing in background checks. I'm sure they have a policy on this, as providing background credentials is literally the primary function that colleges serve once you graduate. (Compare the number of alums to the number of enrolled students at a typical college: undergrad institutions' core business is not teaching, but vouching for people they once taught)