I've thought about it a little more and here's my latest idea:
Use a router to inset the mounting plate for the Ultrastik 360 about 1/4 inch into the top of the control panel. Then, route out a 'channel for the restrictor plate above that. Imagine if the 4-way restrictor was split into two halves, each half being a 1/8 inch thin plate on either side of the stick, embedded in the routed channel. To restrict to 4 way, slide each half in until they meet, joining around the shaft to form the diamond of a 4-way restrictor. To de-restrict, slide each of them outwards. Imagine a further thin layer on top of this, so that the restrictor was not visible to the naked eye, and have a layer of plexiglass on top of that. The restrictor plates woud have a peg coming up out of the plexiglass via a channel that allowed you to slide the restrictor plate halves into and out of position, and to lock them into position as well.
For bonus points, imagine several layers of this: one 4-way restrictor layer, one 2-way horizontal restrictor layer, and one 2-way vertical restrictor layer.
All of this requires pretty thin restrictor plates though, otherwise you start having to put the mounting plate of the stick down too far to fit all of those restrictor plates on top of it. A long-handled shaft would help in this regard. At some point, it might even be worthwhile to consider bottom-mounting it with a long shaft which would give even more room for restrictor plates embedded in channels in the control panel.