Well, I didn't have a 15/16 socket, or anything else that might have functioned (I had gotten out the dremel and everything) so I decided to take the needlenose pliers and a channel locks route.
My needlenose pliers were nice and fat, and fit in the notches rather well. I squeezed as hard as I dared, then grasped the open jaws with the channel locks and it worked fine. Spun it off, put on the new plates, spun it on, tightened everything up, and it appears to work.
My old clutch plates were ~.10 inch, the new clutch plates were ~.14 inch, so there was a difference, just not much. The new ones are definitely fatter, and when I put them all together, the end result was - at least to me - "wider".
It's 10:16, and I have a toddler in bed and a wife on the way home so it's too late to take things for a test drive tonight, but tomorrow for sure! I put on a new sprocket too (1 tooth larger) so we'll see how she goes!
Thanks for the help, everyone!