I went back and redesigned a home. When I was done I took a new photo during the completion sequence. However, Happy Home Network still uses the old photo. Can I change it to the new photo so that it reflects the changes I made?
After doing some more research I've concluded that this most likely is a bug. I went back to the home a third time and took a photo (not as nice as the photo I took during the completion sequence though) and now I had the option to change. The "middle" photo wasn't available though (event though it's in the album on the Switch main menu).