I intend to make this toy out of plywood and wooden dowel pins. A combination of the Makerworkshop being closed for the week and the desire to get some peer review feedback before building the final version led me to fabricate a works-like prototype from cardboard.
One potential issue that this mock-up highlighted to me was the undesirable out-of-plane tipping caused by the additional weight of the fixtured object moving the system’s center of mass forward. In this configuration, the object is essentially held on by friction against the pins, which is non-ideal. The magnitude of this effect will be smaller in the final version since the fixtured object will have a much smaller mass relative to the plywood board. I think I will be able to correct this tipping in the final version by calculating the moment contributions of the board and the fixtured object, and offsetting the screw eye attachment point backwards.