Add extra (dropdown) field to gdm3 greeter

I'm really going far outside my comfortzone here, but I was wondering: can I add a third (dropdown) field to the gdm3-(GTK)-greeter and then capture the input that it sends (probably to the Gnome session)?

Usecase: I have several hotseats in my office, with a VoIP-telephone next to it. I have configured a click-to-dial add-on in firefox (telify), but of course, when you're not at your 'regular' desk, then this is worthless. Thus I would like to add a dropdown select, giving the users the possibility to select which telephone is on their desk.

Is this possible? Thx for any pointers!