After much digging, I discovered that while the reference manual for STM32F4xx says you can have 3 IN endpoints, it fails to prominently specify that these can only be the endpoints numbered 1, 2 or 3.
The HAL library code allows the use of any endpoint number from 1 to 15. I was able to choose endpoint 6 in my code and it compiled and executed without complaint, though, obviously endpoint 6 remained silent.
Further digging revealed this caveat in the PCD Initialization Structure:
I hope that my failure to R The FM might save someone else from making the same mistake that I did.
Unfortunately, my application needs to use endpoint 6 in order to match an installed base of drivers. I don't see a way to do that using STM32F4 parts, but if someone can show me a way, that would make my day.