[sbierwagen]

I don't know if there's any strain gauges designed for this application. I'd be worried about nonlinearity, creep and how you would calibrate it in the field. (If it requires a dial indicator and a surface plate, you'd probably be way better off getting a regular CNC mill)

Optical tracking might be easier.

I'm not sure how much deflection you could actually tolerate before deforming the joints in the arm. Even with deflection tracking you might end up taking very, very shallow cuts. A 3D printer that runs for 16 hours in your apartment is annoying, a CNC mill running for 16 hours will be genuinely intolerable.