Flying ERJ or CRJ CAT III by hand on the HGS has nothing to do whatsoever, one or two engines on, with the indication Cat III single/dual on an A320 !!!
The Airbus indication has to do with the Autopilot being fail/operational or fail/passive, as there is no autopilot doing CAT III for you on the ERJ CRJ the two don't compare...
Nic