As the dense deployment of access points (APs) in cell-free massive multiple-input multiple-output (CF-mMIMO) systems presents significant challenges, per-AP coverage can be expanded using large-scale antenna arrays (LAAs). However, this approach incurs high implementation costs and substantial fronthaul demands due to the need for dedicated RF chains for all antennas. To address these challenges, we propose a hybrid beamforming framework that integrates wave-domain beamforming via stacked intelligent metasurfaces (SIM) with conventional digital processing. By dynamically manipulating electromagnetic waves, SIM-equipped APs enhance beamforming gains while significantly reducing RF chain requirements. We formulate a joint optimization problem for digital and wave-domain beamforming along with fronthaul compression to maximize the weighted sum-rate for both uplink and downlink transmission under finite-capacity fronthaul constraints. Given the high dimensionality and non-convexity of the problem, we develop alternating optimization-based algorithms that iteratively optimize digital and wave-domain variables. Numerical results demonstrate that the proposed hybrid schemes outperform conventional hybrid schemes, that rely on randomly set wave-domain beamformers or restrict digital beamforming to simple power control. Moreover, the proposed scheme employing sufficiently deep SIMs achieves near fully-digital performance with fewer RF chains in the high signal-to-noise ratios regime.