I have an update here.
I kept playing around, and found that small batches were working for me, but not when I was trying to update the whole platform. I set out to investigate, and what I found was what appears to be an issue in handling duplicates. This is a little strange, since I have duplicates for other platforms that haven't exhibited any issues. However, I confirmed through various tests that when I selected batches that had multi-CD games that identified as the same game in LaunchBox, the movie download portion was not working. Regardless of batch size, when I selected a batch that did not have any duplicate-identified games, the movie downloads would work as expected.
I haven't sniffed the wire to dig deeper yet since I've already resolved my issue, but based on how the UI is behaving, it would seem that the issue is in LB querying EmuMovies with a duplicate title and EmuMovies not liking that very much.