I believe your tray will accomodate the Nauticam housing but am not positive. If you had the TR-D tray it would not work because it is designed to mount with a single screw. I think it is likely you would need to add the tr-dup or tr-dupl to the TR-DM so it is long enough. Or you could get the TR-IK or TR-SBLD trays which are longer. I think I am using the TR-IK and TR-DUP with my Nauticam EM-5 housing, but would have to check to be sure. Even if you need one of those a longer trays you most likely could use one of both of your current handles, keeping your cost pretty low. The TR-IK tray is only about $50.00 new and if you are not in a hurry I think you could find a used one.
The Nauticam housing will work not only with its own tray and handles, but also with ULCS stuff as well as trays and arms from a number of vendors. For me, I found the ULCS tray to work better for me than the Nauticam.