After an extended period of round-and-round with Netatmo tech support, I ended up purchasing a replacement outdoor unit for our weather station. So, we once again have a fully-functional installation. The process of getting to this point bears mention.
Netatmo support is glacially slow. It took over a week to get a response to an email or contact initiated via their website. Further, their responses are less than enlightening. After several messages back and forth, over a period of weeks, I resorted to public outreach via Twitter, referencing the ticket number. This at least seemed to accelerate their responses.
In the end, since the device was long out-of-warranty, all that I could do was purchase a replacement. I could not find a replacement available from common online sources like Amazon.com. The indoor modules are offered, since you can add more of those to address other indoor spaces. The system only allows a single outdoor module. I infer that makes replacing outdoor modules somewhat uncommon.
I eventually found it offered on their website for $69. This was much less costly than an entirely new system from a different manufacturer.
Now that the weather station is working correctly, I can revisit integration with with our local instance of Home Assistant.