As this has progressed enough to at least be functional, I've decided to publish it. The videos and pictures here are of my "V1" of the project. I only need to install the battery and charging circuit to consider this version complete.

Testing has shown this can generate 150-170g of pressure at full reach, which is reasonable as the cheap 9g servo I'm using is supposed to generate 1.8kg/cm of torque, and the distance between the servo's pivot and where it was pressing on the scales is about 10cm. The problem is that REX buttons are specified as taking anywhere between 200g and 1.5kg (expressed as approximately 2-15N) of pressure to trigger, though the one's I've observed have all been at the lower end.

V2 then really has to target 300g/3N to be considered worthwhile, which will require a servo that's more than twice as powerful. I'm considering this one as a candidate: https://core-electronics.com.au/micro-servo-high-powered-high-torque-metal-gear.html with an estimated 3.2kg/cm at the 5V I'll be working at.

Additionally, V2 will be built using a 3D-printed 4-bar linkage so that this project is repeatable, and all the files and code will be published at that point.