Interesting problem. I think I follow what you're trying to do.
I modeled a big cube on the far wall. I restrained it to move along the axis of the cone. I made it rigid, which means it can't deform. Making it rigid also makes SW mesh the exterior as a shell, which means its not very computationally expensive.
I put a spring between the cube and the cone face. I made the 'normal stiffness' super high, so the cone face can't move toward or away from the cube. The tangential stiffness is 0, so it can slide (translate) freely.
I used a displacement plot to help verify the constraints are working. The plot is for UX, the direction parallel to the axis of the cone. I set the min and max values to bracket the values shown on the cone face edge. The uniform color indicates they are all moving the same distance, hence they are moving together. They are non-0, indicating the face is indeed translating.
I think you're making a quarter section of the cone; you might use the symmetric fixture instead of 'on plane'. They are equivalent, but with the symmetric fixture you can mirror results.
Thanks a lot for your very extended answer, it helped a lot!!!