I figured it out!
Well, partially. The dispersion nut isn't held in there by anything other than coffee residue. Taking apart the Program V I just bought, the nut fell out with the shower screen. Sure enough, I just tapped this one with my screwdriver and out it came!
Does anyone make a dowel spanner that will fit the dispersion plate? Or do I have to fabricate my own? I can't find any service tools on the NS price list.