Cut Geometry with a Solid Form

You’ve got an awful lot of geometry to deal with. It wouldn’t hurt to simplify the rock. Turn on Hidden Geometry temporarily so you can be surgical about the geometry you select and use Intersect Faces>With Selection. Then turn off Hidden Geometry so you can select and delete surfaces.

It would also help if you convert the rock to a component and then use the Dave Method. That will help you avoid the tiny faces issue.