Success!
I've downloaded the dropbear source code on my desktop, built it, and now I have the conversion tool, dropbearconvert.
This has allowed me to convert my existing openSSH key to dropbear format.
However, that step was not needed!
All I needed to do was to create the authorized_keys file on my dream, just as I would have created it on any other computer, containing the public key of my desktop (host where my sessions will start from), and deploy this file into /var/.ssh directory.
How did I know that was where I should put it? Simply checked the /var/scripts/dropbear_script.sh, the script used by enigma to control the dropbear daemon, and noticed that this script ensures the existence of /var/.ssh directory (if it doesn't exist, create it).
My success was partial because it only worked on 2 out of my 3 dreamboxes. Funnily enough, it worked on the DM7000 and one of the DM500s, but not on the other DM500, where I'm still prompted for a password. All boxes are running the latest ****** image, so the two 500s should literally be identical.
Case closed!