Can anyone tell me what this thing is out of a Cat C-12?

That piece is the bypass regulator diaphragm for a Caterpillar C12. It sits down in the thermostat housing under the thermostats, right in the center of that bore.
When you put it back in the little raised sleeve in the middle goes down toward the engine. The flatter rubber side faces up toward the thermostats. The outer edge just presses into the recess in the housing and needs to sit flat all the way around before you drop the thermostats back in.

Its to control the internal bypass so when the thermostats open it forces coolant through the radiator instead of just circulating inside the engine. Just make sure it’s seated evenly and not ####ed sideways before you bolt the housing back on.