Hi Elliot, when you extrude a curve you'll only get a solid result with capped ends if your curve is closed and planar.
But if you extrude a surface you'll be able to get end caps in the situation you're describing. So the usual method would be to trim the surface with your curve as a cutting object, and Trim actually has projection built into it so you don't need to project as a separate step. Then when you have your surface fragment you can extrude that into a solid.
