I have an DRM-604X next to my Amiga and it´s normal that it goes through all the disks.
What I had to do (or wanted) was that I made Dosdrivers for all of the 6 disks that pointed at 6 mountlists... iirc that is. Then I mounted everyone and use the diskchange command to see the icons from the disks, but you can only have one "starticon" at the time because it´s only one disk at a time.
Did that make any sense or should I go and sleep now? :-)
I´ll take a look at it when I get my Amiga working again, new cooler is required for my A3640.