Sandbox could do it but you’d need to turn the box to vertical and put it under the shape you want to project.
What are you really trying to accomplish, though? Maybe you just need to rotate the shape and use Push/Pull to extrude it through the side of the box. Then use Intersect Faces.
If you share the .skp file and give us more information it would help us help you.
You still haven’t explained exactly what you are trying to accomplish. Is what @Box showed thething you are after?
Here I’ve projected the shape through the side of the box, then selected the outer surface only and used Intersect Faces. Then I used Joint Push/Pull to extrude the region inside because you wrote:
Think of projected as what you’d get with a slide in a slide projector. In this case the result that Box and I show is projected. Draped would be what you get if you dropped a thin pancake onto the curved surface like Miha.s shows.
Hope you had a great Thanksgiving.
Thank you so much for clarifying.
I had tried an extension by Didier Bur called projection extension about a month ago. I found it easier to just stay with intersect selection (as you showed). Intersect is my bread and butter for projection and Trim for cutting.