Oh, so that's what "Use by 4/10" means.
I had some nagging doubts about the silicone I applied last night on the lens and eye, and I was right to have them. This morning the silicone hadn't cured, so on the way home I picked up a new tube. When I got home, I finally checked the expiration date on the old tube, and it read as I wrote above. Note to self, and anyone else using GE Silicone II: Beware the expiration date.
I popped the lens back out without a fight.
After thoroughly cleaning the lens and eye, I used the new tube (with an expiration date of 5/12) to reglue the lens. This tube of silicone behaved much better - the bead was thinner, more viscous, and easier to apply. I think it'll work this time. Maybe I'll take up decorating cakes next.
I smoothed out the silicone, and I'll hope for the best.