Use the _Trim and _Split commands.
You can try to explode the surface using the _Explode command. If you do this, you will need to pick each individual surface to cut.
There is more than one path to get here. The answer depends on your end-goal. These solid tools are going to handle all this. Take some time to understand how these tools work and what they need to be successful.
I am still willing to bet rebuilding both of these polysurfaces will yield the best results. Take polysuface 1 from your example: If you explode it, you’ll get 1,185 individual trimmed surfaces. You can get this down to less than a dozen while still maintaining the integrity of your shape.
The Help Tab is reeeeealy good for explaining how these tools work step-by-step.