I'm happy to report there's now two working doorbells on our lovely little home. I decided on a different style than the front door's new bell. Price was the main consideration, I'll admit. But I think it's also fitting to have something a little less flashy on the back door. The black also fits nicely with the black door.
I'm not sure who'll ring the doorbell from our back yard, but it'll be ready if and when someone does!