A FR door to a cupboard as described provides separation from whatever the contents are to the staircase and vice versa. Who knows what someone has put in there; perhaps cleaning products or materials that do not like to be mixed. Conversely, a fire started in the staircase by say, a pram against a wall heater, would be prevented from involving the contents of said cupboard.
That dealt with; if there is detection in the staircase but not in the cupboard, would it not be the correct course of action to fit intumescent strips only (i.e. without cold smoke seals) to the cupboard door in order to allow a little smoke to pass the door to activate the detector in the staircase? In any case, once the intumescent strips have done their job, a fire within a cupboard would surely very quickly run out of oxygen.
Doesn't BS5839-1 only recommend detection in cupboards over 1m2 in an L1/P1 system (unless there's an arson risk)?