I've never implemented it, but I researched using Ethernet/IP to get Fanuc error messages for a customer several years ago. I don't have any experience with the Rexroth system, so I'm not sure what is possible on that side of things.
The Fanuc Ethernet/IP Setup and Operations Manual (MAROC77EN01101E Rev H) has the information you're looking for.
You have to use explicit messaging, but you can get what appears to be all of the information that is displayed on the teach pendant.
You can get the last 5 active alarms or the last 5 alarms in the history. Each of the 5 alarms needs its own message or messages depending on which data you want to transfer. There is a total of 228 bytes of available data for each alarm, but you can pick and choose. They provide everything from alarm code to the readable messages.
My access to Fanuc's documentation is on their password protected website, so I can't send you the document, but I can help you through the explicit messaging parameters, if needed.