Extruding to subtract non-normal from plane

In simplest terms, I have a cube. I also have a shape that is fairly complex and cylindrical.

I want this cylindrical shape to be at an angle, say 20 degrees. Then I want to cut down in the cube from one surface to the item.

This would have a half cylinder bottom with straight sides to the top.

So far I can subtract the cylinder shape from the cube and projected the cylinder to a plane at 20 degrees. I hoped to extrude (or subtract) into the cube down to the widest radius of the cylinder to slice that out. I can’t figure out how to extrude non-normal to the plane.

Second try was to copy the shape up and try to figure out how to make it a 3d shape that I can subtract, but I can’t figure out how to join two sketches together to make a 3d shape.

Anyone have a solution for this?

I can almost get here by projecting to flat plane. Extruding down and then slicing at the angle, to join with the angled cylinder at the widest radius. However, this doesn’t work for the curved edges of the cylinder for the bottom half of the cylinder.