How about this?
Two cylinders, internally seeded with weights, rotating. Between them is a smaller cylinder with a strain gauge, like the ones used in early gravity-wave experiments. The shafts rotating the cylinders could be piano wire or something, so they don't communicate any information about the weight locations to the support structure.
Rotate this for a few weeks, signal average, do some math.
Maybe make the central thing a dumbell shape, to make the strain gauge more sensitive. Use a rubber-band suspension, like an AFM.
I wonder if you could measure G using really high frequency motion, maybe with a mechanically resonant detector. There's potentially more signal energy if you're flailing mass around fast.