The Troubleshooting tool passes. It's able to connect to the iDRAC7 and shows me all the "profiles found on the remote device".
In the OME UI, all the servers are classified in the group "RAC".
I was able to get one of the 2 troubled machines inventoried by changing the timeout from 15 seconds to 60 seconds. But this one machine is being very stubborn. I also upgraded the iDRAC firmware to the latest 1.37.35.