Ok, I tested it and unfortunately it doesn't work.
I setup an AD account named "miteladmin" with a very strong password and granted it full rights to my mailbox in exchange. I then changed my mailbox in the 5000 to enhanced integration, entered the miteladmin username and password, but left my email.
I received the 'welcome to email synchronization' email from the controller which was promising, but the voicemails don't forward.
In hindsight, the controller is obviously using the username to open the mailbox like one would when logging into OWA and so this idea would never work. It's a shame that this is the case, as it means we will be leaving our org on forward & copy instead.
Way back when I was running an Exchange 2000 environment we ran BES (Blackberry Enterprise Server) against it, and it used the scheme I proposed in my original post. You would create a BESAdmin account, grant it rights across the exchange org, and then it would handle message administration across all the users. I believe it still works that way (although I've been lucky enough to not have to touch a blackberry in years.) I was hoping for a similar scheme within the Mitel environment, but it looks like today isn't my day for it.
Thanks for everyone's input regardless, it is appreciated.