on the AMC site, clear the hardware ID. Select the option to replace the controller.
Now synchronize the EX controller to the ARID. It will pull the licenses assigned to the app record. I have done this in the past LX to MXe, MX to CX, etc.
When changing controller types, I will clean their database on a surrogate controller of the same type they started on.
As for restoring the database - use the migration tool to convert it - I have never done this, so I don't know the exact process.