I'm not sure if this got no replies because the solution is considered too obvious, but it did take me a couple of days hunting around and experimenting before I figured out how to do it. And I'm not absolutely certain that what I've done is actually "correct", but it works.
I'll put it on here for anyone else that's searching for the same thing.
It took me a while to work out where the tabgroup was. Turns out it's stored in the Zimlet dialog object. So, my Zimlet is mailaggr, and the dialog is _aggrDialog. Using Firebug, I can see the existing tabgroup by entering mailaggr._aggrDialog._tabGroup into the console.
So, it's just a case of adding the two fields to this tabgroup (and setting focus on the first field):
The tab key now cycles focus from username > password > OK > Cancel.
I hope that makes sense to anyone else that's not a Zimbra expert and is poking around trying to work this stuff out.