How do they put the charge through so as to bypass Rewards Network's "cut"? Whenever I've had to put through a missing visit request, the reps have told me that their office "investigates" the failure-to-credit.

You had better believe I send in puny receipts for under $5 that fail! One restaurant here takes 5 - 6 days to post; most transactions go through on the fourth day after the visit. On rare occasion, I've seen a post on the third day