I turned on Friction and made some other adjustments, but there is a point in the Static Structural simulation where a dynamic event occurs.

In one increment, the material is here:

In the next increment the rubber has jumped to the left, releasing a lot of strain energy, and the solver cannot find Static Equilibrium again and stops.

The corrective action is to solve this in a Transient Structural or Explicit Dynamics solver.

There is a method you can enable the Static Structural solver called semi-implicit to automatically switch to the explicit solver to solve the dynamic part of the solution and later to switch back to Static Structural. That might be worth trying here, but note that the material should have a density material property because there must be mass to compute a dynamic solution.

The small cylindrical face on the rubber part at the tip may be contributing to the dynamic event. If the conical surface were to continue all the way to the flat top of the rubber, with a small blend on the corner, that might improve the solution.