First, we must remember that Room's exist on the phase they were placed on. That is to say, whatever phase your view is set to when you place a room, that is the room's phase.
Build your existing building and add all the room names in the Existing Phase.
Now that you have all three phases set up and views for each phase. Everything that you demolish should be demolished on the "Demolition" Phase... Not the "New Construction" Phase. This is where my mind set had to change a bit. I had always set the "Phase Demolished" to "New Construction" and I think it was due to the Revit Template only started with two phases. **Note** When using the Demolition "Hammer" from the tool bar make sure you are in the Demolition phase view. The hammer sets the elements demo phase to whatever view you are in. ***
Finally, in your Demo floor plans... Set the phase to "Demolition" and set the phase filter to "Show Previous + Demo"... Go back to your Existing floor plan with the room names and select all the Rooms... (Not the Room Names... you want to select the blue squares). Copy to your clipboard, and paste the Rooms into your demolition view.
Tah dah! Now you have your Existing Room names in your Demo views...
Of course, the one caveat is that you are managing two sets of Existing Rooms.