I tried changing the Internal Testers option, tried changing other items on the app details page, and tried uploading several new binaries (with the same code, only changing the build number) over a 2-day period to no effect. I also contacted Apple through the iTunes Connect support link; they replied in less than 24 hours, but only to ask for more info (most of which I had already provided).
Finally, the fifth build I uploaded, two days later, went through in the normal few minutes and I was finally able to submit my app for review.