We present a new scheme for the simulation of the barotropic Euler equation in low Mach regimes. The method uses two main ingredients. First, the system is treated with a suitable time splitting strategy, directly inspired from [J. Haack, S. Jin, J.-G. Liu, Comm. Comput. Phys., 12 (2012) 955-980], that separates low and fast waves. Second, we adapt a numerical scheme where the discrete densities and velocities are stored on staggered grids, in the spirit of MAC methods, and with numerical fluxes derived form the kinetic approach. We bring out the main properties of the scheme in terms of consistency, stability, and asymptotic behaviour, and we present a series of numerical experiments to validate the method.