Wow, tough application. Your measured object's "surface" is variable, inconsistent, unstable, and resilient.
I would consider one of the laser-scanning displacement sensors marketed by Keyence. Fabulous devices, but expensive. Perhaps the other major sensor manufacturers have some clever devices also: OMRON, SUNX, Banner Engineering, SICK (as mentioned previously), Balluff, probably a dozen more that I could think of on a good day.
If your object has some resistance to it, then perhaps you could rig up a contact probe sensor of some sort. Keyence makes those, too. Or perhaps some sort of simple limit switch affair with a "paddle" attached to the swing arm to distribute the switch's imparted load on the squooshy surface across a larger area.
But one thing you MUST do is a better job of specifying your needs. The specification "with average precision" has no merit.
TygerDawg
Blue Technik LLC
Virtuoso Robotics Engineering