Turned out it was mount. I fixed that and haven't had a problem since.
It was also, to a much smaller degree, the shell I was shooting, and compounded by the fact that I was shooting from the bench. I don't have a lead sled, so I was taking all of the recoil myself (in the wrong spot, mind you) and I couldn't rock back with it like I can when I'm standing or kneeling free. The shell was a 1-7/8oz #5 turkey load rated at 1500 ft/sec. MV (I think). So a high recoil shell, a bench rest, and mounting in the wrong spot. No wonder I wasn't getting the patterns I wanted.
I fixed the mount and the rest fell into line behind it. Borrowed someone's lead sled and got the patterns I wanted out of both the 1100 and the Rem 870 (turned out to be different rounds in each, which isn't unexpected). Shot several at the range like I would in the turkey woods with both guns, and bingo. No pain. Both of them still kick like crazy compared to the #7 bird shot I shoot varmints with, but it's very managable. I'm especially impressed with the 870.
ETA: I'm still going to get a new pad for the 870. I'm thinking Limbsaver, but I've handled some of the Remington SuperCell pads and I like them. I may actually try that first.