Now what do I need to do next to discover a means to access the data, please?
I don't know that anyone here has the experience with a Linux project such as you've laid out, to be of much help.
We can tell you how to do it with Windows, and even some Mac people hang out here.
Based on my experiences, though, I would recommend breaking the project up into small chunks and make each chunk work before moving on. Else you will never know what's wrong if it doesn't work.
For instance, it seems as if you want to connect to a serial-over-ethernet adapter. I would install the TCP-IP com port software and make sure that it can talk to the adapter and then to the console before moving on.
Then you could install the demo Linux version of Weather Display and make sure that it can actually talk to the console and download data.
If you are having trouble with the Argent device, there may be a problem with handshaking line. Only 7 and 8 are jumpered in the DSI-01 green plug.
Keep us posted. You're breaking new ground here.