I was able to finally get Google Photos to back everything up. It's not a perfect solution, but it does seem to work for now.
Uninstall / Reinstall.
I just completely uninstalled the app, then immediately reinstalled it from the Google Play store. One of the first things it asked about in its "first-time setup" procedure was whether or not I wanted it to backup my files over cellular data. It defaults to "Wi-Fi only", but I changed it so it would back up on both Wi-Fi and cellular data.
After going through the rest of the "first-time setup" prompts, it began backing everything up, as well as giving me a "clean up ### MB of files" option. I chose that because, why not? I noticed that it did go ahead and back up all of my photos, but the spinner kept running at the top of the app, saying 285 items left (I'm not sure of the exact number) and appeared to be stuck at that.
I decided to let it go and see if it would eventually get through them all anyway. Hours later, I saw that it was on 162 items left. Finally, hours after that, I got the much-sought-after cloud-with-a-checkmark icon. It had finished backing them all up.
Perhaps it got stuck on some long video files or burst-shot photos—I don't know. In any case, I think the uninstall / reinstall procedure is what did the trick for me.