The interaction between surface acoustic waves (SAWs) and spin waves (SWs) in a piezoelectric/magnetic thin film heterostructure yields potential for the realization of novel microwave devices and applications in magnonics. In the present work, we investigate the SAW-SW interaction in a Pt/Co(2 nm)/Ru(0.85 nm)/Co(4 nm)/Pt synthetic antiferromagnet (SAF) composed of two ferromagnetic layers with different thicknesses separated by a thin nonmagnetic Ru spacer layer. Because of the combined presence of interfacial Dzyaloshinskii-Moriya interaction (iDMI) and interlayer dipolar coupling fields, the optical SW mode shows a large nondegenerate dispersion relation for oppositely propagating SWs. Due to SAW-SW interaction, we observe nonreciprocal SAW transmission in the piezoelectric/SAF hybrid device. The equilibrium magnetization directions of both Co layers are manipulated by an external magnetic field to set a ferromagnetic, canted, or antiferromagnetic configuration. This has a strong impact on the SW dispersion, its nonreciprocity, and SAW-SW interaction. The experimental results are in agreement with a phenomenological SAW-SW interaction model, which considers the interlayer exchange coupling, iDMI, and interlayer dipolar coupling fields of the SWs.