I would also add the following - although not directly as to whether the code requires the protection. (I do agree with Handler's initial response, by the way.)
Consider that your primary objective is to serve your client. Without the rated construction your client has an increased exposure to damage and/or physical injury or death in either of two circumstances: a) a fire in the adjoining building cheek-to-cheek with your new deck/stair spreading to your building (it does not take long for wood to 'go up' when there is zero separation from a source); b) a fire in your own building that spreads to the adjacent structure, causing damage/injury/death and not covered by insurance because the carrier will argue the design was deficient (the spread of fire was both foreseeable and predictable); c) a fire involving both structures renders the new exit stair unusable due to flames and/or smoke, preventing the occupants in your building from reaching safety.
Even if the City for some reason allowed you to build without a rated deck/stair (which I doubt), I would encourage you to discuss the practical impacts with your client... and I would at a minimum suggest using non-combustible materials for the entire new work.
An insurer may also not want to take on the risk of this building with the new non-rated construction that presents an increased risk - or might increase premiums for coverage.
Whether to proactively seek safer solutions or not is a question that can go beyond what the code minimum requires - and sometimes requires that you go to battle with your own client (in his/her own best interests, as well as yours, even though initial construction costs may be greater).
I hope you get it straightened out with the BO, but would also encourage sitting with your own client and discussing merits and options.