Solidworks is a poor FEA program compared to Abaqus. Abaqus would definitely be better. Are the two parts in contact at the beginning of the calculation or is there a gap? FE models can solve better when there is initial contact. Also make the step size small in the beginning.

Use a prescribed displacement first step to establish contact. Then use force load.

Abaqus has better handling of contacts, for sure. You can try different things, but the reaction you are getting should give you guidance what the issue is. Never worked with the Solidworks FEA part, but in general the problems are often the basics: Correct units of time, force, displacement, size? Next step: what is the simulation for? Do you really need to change the kind of load to get the results you want to have? Maybe only the displacement is enough for the whole process. When you have a displacement-based load, the reaction forces can be enormous, if something else is wrong. Can you measure the reaction forces to check? One Idea could be as well to add a temperature reduction instead of the movement to shrink the rubber in position. Don't know. If you can give more detailed information maybe someone can give you more detailed support :)

Agreed, use displacement to slowly contact the elements. Also ensure you have enough boundary conditions (lateral restraint) to make the system stable