For anyone that is following along, or sees this problem themselves.
We found an message in the logs saying the Collection that the Product was associated with had an invalid layout. Perhaps the layout's version was updated or something like that - I'm not sure. That error prevented the Product from being saved. Assigning the collection another layout, saving, and then assigning the original layout cleared that error and the Product saved properly.
When I have a chance, I'll look into providing a more informative message that helps the user quickly diagnosis the problem themselves.