Mark: I don't know about LA's product approval process, but sprayed polyurethane foam insulation meets the requirements for flame spread per Chapter 26.
Additionally, the intumescent ignition barrier prevents flame and temperature from reaching the foam like it does for steel.
And, yes, the foam will reduce headroom height, but so will any insulation type added to the underside of the slab. Polyurethane foam has an R-value per inch that is higher than any other type of insulation on the market; so using another product will reduce the headroom even that much more.
Insulation could be added to the top-side of the concrete deck, but it should have a fairly high compressive strength (like that used under plaza deck pavers), to withstand the loading with minimal deflection.