A grating-based x-ray multi-contrast imaging system integrates a source grating G0, a diffraction grating G1, and an analyzer grating G2 into a conventional x-ray imaging system to generate images with three contrast mechanisms: absorption contrast, differential phase contrast, and dark field contrast. To facilitate the potential translation of this multi-contrast imaging system into a clinical setting, our group has developed several single-shot data acquisition methods to eliminate the necessity of the time-consuming phase stepping procedure. These methods have enabled us to acquire multi-contrast images with the same data acquisition time currently used for absorption imaging. One of the proposed methods is the use a staggered G2 grating. In this work, we propose to incorporate this staggered G2 grating into a state-of-the-art breast tomosynthesis imaging system to generate tomosynthesis images with three contrast mechanisms. The introduction of this staggered G2 grating will reject scatter and thus improve image contrast at the detector plane, but it will also absorb some x-ray photons reaching detector, thus increasing noise and reducing the contrast to noise ratio (CNR). Therefore, a key technical question is whether the CNR and dose efficiency can be maintained for absorption imaging after the introduction of this staggered G2 grating. In this paper, both the CNR and scatter-to-primary ratio (SPR) of absorption imaging were investigated with Monte Carlo simulations for a variety of staggered G2 grating designs.