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Those are valid limitations, but current 3d imaging techniques I know of are either with photography, laser scanning or touch probing. All of the above also face those limitations.

My initial interest was that it would bring the cost down a lot, but they don't seem to bring great precision and they require a moving robot with at least 3 axes which is unlikely to be very cheap for precision material.

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You don't need a robot arm at all. Consider clamping the test object inside a transparent sphere, resting on wheels that allow the sphere to be rotated. Instead of dipping, partly fill the sphere with liquid, test displacement at many orientations, then return to upright where more liquid can be added, repeat. You'd get the same measurements, just in a different order.

The robot arm is expensive because it needs precision hardware (backlash free bearings/gearing).

Your idea also requires precision hardware to move the sphere.

Actually besides 3 step motors ( for rotating in three axis ), rubber wheels to hold the sphere and a sensor to measure the water level, I don't think you need more. What I don't know is how much water you need, probably this can also be computed.

It's a better idea than dipping with a robot arm. And also faster.

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