Update for today: I got home last night and attempted to unpair the watch. I received an error saying the unpairing was unsuccessful but I still got the pairing screen on my phone. The watch also reset after a short time.
I went through the pairing process and restored from my backup. After getting everything set up I went to the software update screen and sure enough, there was 4.2.2. I hit the update button and the progress bar briefly showed up with "Downloading..." then that was replaced by "Estimating time remaining...". A couple of seconds later the update screen was replaced by the all-too-familiar "watchOS 4.2 - Your software is up to date." screen.
So I'm giving up. I don't know if I should schedule a genius bar appointment for this since my watch is working fine where it's at and I really don't want to take the chance of losing all of my activity data since it seems like I'm going to have to set up as new to get around this.