Well this worked, but it was kind of finicky.
I made a rectangular tab, using the dimensions to make sure that the tab was 1/6th the height of my path. Copied this out 6 times, grouped the tabs, and copy/pasted the group so that there were tabs on either side of my path. Aligning the tabs perfectly was a real pain, but eventually I got it.
Then I deleted every other tab.
And last, I was able to select all the tabs and my original path and use the Pathefinder->Exclude tool to produce the shape I wanted.
That's about as efficient as I can think of. The biggest time sink was in the alignment. In order for these to fit together they really need to be perfect. I think the reason it took so long was a combination of my inexperience and poor snapping settings in IA.