Good point on "moving" the files instead of deleting.
I stopped WC, created a temporary directory and moved the two existing WC directories there. Then I restarted WC, confirmed that WC generated two new WC directories WeatherCatBackups and WeatherCatData, and WC did not modify the timestamp of the files in the renamed directories. I then looked at the Pref pane in the WC client, and all of my WC pref settings for WU and CWOP were preserved. It is almost like WC has another place where this data is stored (not in the plist files in the WeatherCatData directory). Or this data is cached somewhere somehow, not sure. Once again, I confirmed that CWOP was running and uploading data as expected, but there was nothing about WU anywhere in the log/console.
The strange thing is when I enable the WU option "Log Historical Uploads", the date of the uploads WC is looking for (according to the log) is October 18, 2020, which is about the time that I stopped WC and took down the weather station prior to making the move. Perhaps there is a WC bug that prevents the proper starting of the WU service if the previous Pref setting for WU was set for Enable Historical Upload. Possibly, WC is trying to upload that historical data before it will start properly. The fact that I changed the name of the Station ID might have something to do with it. When I took the station down before moving, I removed the battery and boxed it up until just a day or two ago, so it hasn't been recording any data since I made the move in October.