The clamp method, I'm happy to report, is a rousing success. Here's the technique I used. I first seated the leather cover's plastic channel at one spot on the base's edge and secured it with a beefy clamp. Then ten or fifteen minutes later I added another clamp about an inch further down. And then another an inch down from the last one. I proceeded in that manner all the way around the base's edge. A clamp can exert far more continuous force than any human hand. You can let them squeeze all night, which is what I did to make the progress you see here. Only one stipulation: You gotta have a lot of clamps, which might involve a trip to Home Depot or Harbor Freight. You can pick them up for a buck or two apiece. You'll want at least a dozen. I had 13 on hand.
This seat has eclampsia. The front was especially stubborn, so I'm going to let the clamps stabilize it overnight.