Tried grease. Didn't work. Tried wet paper towel. Didn't work either, and managed to pack it in to about the consistency of MDF. Went and got a better bearing puller, which worked with something jammed in between the jaws of the puller tight enough that it could not come loose. The back side of the bearing was all fubar from the pressure of the grease/water. (The water squeezed out past the bearing race. That thing was in there TIGHT.)
Clutch is in now, everything is working. Now I've got to wait for the break-in period to end before testing out what the stage 3 can do. The one that was in there before was apparently a stage 2, and couldn't hold a 4000RPM clutch drop. This one ought to be able to do that.