Responding to my own thread, but I believe I've found the answer:
The hardware in the RRC1258 only supports 802.11B. It does not support 802.11G. The issue with WiFi 7 is that they phased out the old WPA-based security for networks and replaced it with requiring a minimum of WPA2. The wireless card in RRC1258 does not support WPA2. It is constantly failing out trying to use old-style WPA connectivity on the new router. I found a bunch of "mismatched authorization packet" errors in the router logs after enabling additional logging to support this conclusion.
As far as I can tell, the only way to connect a RRC1258 to a Wifi 7 network is:
1) Don't run Wifi 7 at all so that you can keep running WPA
2) Run the ethernet jack on RRC1258 to a wireless bridge that supports a later version of the Wifi standard
It isn't a great option, but until and unless RemoteRig releases an updated device that has better Wifi support, this is about all there is.
There are several companies that make Wifi 6 and Wifi 7 bridges, so luckily the needed parts seem to be easily available.