Automating staircase cleaning remains a challenging task due to the confined spatial constraints, complex motion requirements, and the added instability introduced by liquid sloshing in mobile cleaning robots. This paper presents a novel control strategy for a reconfigurable staircase service robot, sTetro-SR, which carries a cleaning liquid and must safely traverse stairs without compromising stability. The key contribution of this work is the development of an optimal output feedback controller that simultaneously achieves trajectory tracking and suppresses sloshing, modelled using simple pendulum dynamics. A decoupled state-space representation of the robot and slosh system is derived, and the control problem is formulated as a constrained dynamic optimisation problem. The proposed solution employs a Kalman filter-based Model Predictive Control (MPC) framework, ensuring that both motion and slosh dynamics remain within safe bounds. Experimental results in real-world staircase scenarios demonstrate that the proposed method significantly outperforms conventional PID control by reducing slosh-induced deviations and enabling precise trajectory tracking. This integrated approach advances the safe and efficient deployment of service robots in complex environments such as staircases.