So first question, you are ok with the problem basically being flow through a tube with an origami plate in the middle? Essentially fixing the piston and changing the fluid parameters around it to simulate the piston motion? If so, the setup sounds fine. This may be a big assumption but beyond it, the problem is outside the scope of solidworks simulation.
On the time dependent analysis, you can get to 3500 iterations with auto time stepping but it stops at a low time value with manual. What is your manual time step and what are your finishing conditions. Sounds to me like you are stopping at max travels. For time dependent, your finishing conditions in your calc control options should be physical time.