Is there a way to apply a clipping plane permanently, like trim down an original plane to a new one based on the clipping plane, to create a new object?
Nothing in native Rhino tools to do that automatically.
It’s not much of a job to create a Cutting Plane for the same object, and use your Cutting Plane to position it. Then trim and you’re all set.
That’s easy enough to do that it’s probably not worth the coding time to write a script.