Your idea of doing a sweep with a constant normal was sound and consistent with the way the blade is probably honed. The problem was first you were trying to do too many things with it-trying to define the boundary, trying to remove material and trying to define the surface.
The fact is, you don't need to control the boundary. You've already established the extents of the blade with your base feature and you're not trying to cut away a specific volume, you want to remove everything on one side of the blade surface, so the blade surface doesn't have to be constrained to the top running edge. It can overshoot or undershoot.
Then, use the face to cut away what you don't want and let the boundary take care of itself.
You can use a 'cut with surface' or 'intersect' feature, but those create boundary edges where your tool surface intersects your base part
That's why that little crescent region is left at the top. (You can clean that up using 'Delete face') but in this case, 'replace face' ignores the tool surface boundary and extends the surface to fit the boundary of the surface you're replacing.
I uploaded the part file (SW15) incase you want to look it over.
