The electrostatic interaction force between a charge-regulated particle covered by an ion-penetrable charged membrane and a rigid charged surface immersed in a mixed (a:b) + (c:b) electrolyte solution is analyzed. The membrane contains both acidic and basic functional groups, and a general form for each is assumed which allows multiple proton transfer among functional groups. The rigid surface is negatively charged, and the net charges carried by the particle is also negative. We show that if the separation distance between the particle and the surface is greater than a critical value, the multivalent cations present in the liquid phase have the effect of reducing the repulsion force between the particle and the surface. The reverse is true if the separation distance is smaller than the critical value. For a constant total number of functional groups in the membrane phase, if the membrane is sufficiently thin, the variation of repulsion force as a function of separation distance exhibits a maximum.