I realize there's a lot of things about this project I'm not doing the best way, but I have to work with what I know in the time I have.
Turns out it was because of my render passes I had set up. There was a background partition set to not let anything in it render. When I merged the scene, the object was put in the background partition and overridden to not render. I moved it out of that partition and it works fine now.
I had tried using the "inspect override" option you get in the object's visibility dialog, but it didn't work. According to tech support it's a documented bug. No word on a fix, though. That would be quite useful if it actually worked. Live and learn, though . . .