It happened that the Warner Brothers play park at Woodfield Mall was outside the Select Comfort store, so I wandered in and was approached by a salesman. I told him that we already had a Sleep Number bed and it was having some problems. I told him about the mattress sides and he said, "You're still under warranty, so I'll tell you how to fix it, and if that doesn't work, here's the card for the service department."
The trick was to unzip the mattress cover, remove the foam from the edge of the bed, and swap it to the other side. It turns out that this does appear to work.
However, daisy_knotwise had previously informed me that the slats in the box spring on my side of the bed were coming loose. So before we performed the unzipping operation, we shoved the mattress to one side (which was the simplest thing to do, because it's tethered to the hoses that keep it filled), and removed my box spring.
It was a train wreck. Two or three of the supports were partially detached and the top cover piece at the head of the bed was partially separated and bowed down. Hmm.
We put the box spring (ok, there are no springs, but it sure is a box) back together and squared it up, then applied a good bit of pressure to bend the top panel back into position and reattach it. Then we reassembled the box spring, fixed up the mattress, and put the whole bed back together.
The box spring did give a couple of loud pops as various pieces settled into place once we got onto it, but it seems stable.
And the bed is now a lot firmer. I guess I need to let a bit of air out...