This paper presents and examines a new design concept for a bistable reciprocating piston pump. The bistable pump mechanism belongs to the bistable mechanisms, which have two stable positions at the end of the suction and discharge strokes. The transition between the stable positions is achieved by using triggering force at each beginning of suction and discharge and subsequent movement using a recuperative spring. In this mechanism, the triggering forces are created by two Shape Memory Alloy (SMA) wires. Geometric and force expressions for the pump suction and discharge strokes are derived. Additional equations are obtained for the balance of moments for the two stable equilibrium positions and the unstable position in the middle of the stroke. Numerical studies have been conducted for the suction and discharge strokes, considering the force exerted by the gas on the piston, which is modelled by an indicator diagram assuming a polytropic process. It was found that the load on the mechanism has significant non-uniformity. The diagrams illustrating the distribution of total moments showed that the cold SMA wire shifted the point of instability. The numerical example shows how to choose the right spring stiffness to obtain energy recovery. In this way, the triggering SMA forces act only at the beginning of the two strokes and, after that, the recuperative forces substitute the action of the SMA forces. The theoretical relationships and methods presented here are suitable for synthesizing new pumps or analyzing similar mechanisms.