Yes, PROJECTGEOMETRY works in such a way.

You can use SOLIDEDIT command to copy faces in which you want to project geometry. (Menu Modify -> Solid Editing -> Copy Faces)

That will create surface, on which you can project geometry and then move the result to original place (you can use Object Snaps and temporary construction lines to duplicate the interposition of solid faces and objects you want to project).

Look at the attached drawing. I copied face (colored in yellow) to new position, projected poly lines on it, then copied the result back.

Or you can simply erase unnecessary results after you project geometry to whole 3D Solid (as I did first time here).