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I want to thank everyone who emailed me with ways of working out the sweep that I was trying to accomplish. I was fairly certain that this was a situation that would call for a varible section sweep would be necessary, and that was indeed the case.
In the end, I ended up need to create 8 features to generate my sweep.
My process (I took the method recommended by Mr. Drum and modified it slightly):
1. Create a surface copy of the outer surface (call this copy A)
2. Create a surface copy of the "lip" edge (call this copy B)
3. Create an offset of copy B a distance upward (into the part) the distance of my feature's width in cross section (copy C)
4. Create a curve of the intersect of copy A and copy B (curve D)
5. Create a curve of the intersect of copy A and copy C (curve E)
6. a. Create a Variable Section Sweep using curve D as the base curve and curve E as the secondary reference.
b. Set curve D tanget to the lip surface at all locations
c. Draw the feature, making sure to extend the shape beyond the surface intersect slightly (necessary to ensure that it will always intersect the surface around the corners)
d. Generate the VSS as a surface
7. Cut the VSS with copy A
8. Solidify the cut VSS
9. Hide (and save visiblity) all surface and curve features that were created