An update to this: this is happening as soon as the 3D switch is flipped (not when it's converted to a shape). As soon as it's made 3D, this piece fills in, as if it was never there in the first place. If the file looks correct in AFX upon import, I don't know what I can change in the Illustrator file. But in case anyone want to take a look, here it is:TOGETHER_allwhite3.ai - Google Drive
Appreciate any help anyone can provide.
You are probably simply exhausting the maximum number of points on the contour paths and the 3D engine can't make sense of your non-convex, degenerate shape. This is nothing all the tweaking in the world will fix. You need to move on to a real 3D program and use booleans.
Update 2- Kept messing with this and got it to work, although the method was not very satisfactory. If anyone opens the above file, you'll see that the missing piece is a closed path - no indication of anything that could cause a problem. Yet in the end, just messing with this path (moving, stretching slightly) was enough to get AFX to preserve it once the 3D switch was flipped. Seem crazy that a problem like this has to be "fixed" this way, and I'd love to hear an explanation of why this happened in the first place. But for now, in case anyone else with this same problem stumbles across this tread, that's the best solution . . . until someone proposes something better.
The bezier interpretation engine in After Effects is different than the one in Illusrrator by just a little bit so occasionally you'll have to wiggle the handle or a vertex just a bit to get things to work. I found that cleaning up corners in AI is usually the best way to handle this kind of problem.