Control of boundary layer instabilities remains a critical challenge in fluid mechanics, particularly regarding the mitigation or amplification of Tollmien-Schlichting (TS) waves which drive the transition to turbulence. This work presents a novel variational formulation to investigate the stability of channel flows interacting with wall-mounted metamaterials. Unlike homogeneous compliant coatings, phononic crystals and metamaterials offer tunable dispersive properties, via Bragg scattering and local resonances, that can be engineered to interact specifically with unstable fluid modes. Recently, these concepts have been applied to phononic subsurface (PSub) architectures, demonstrating potential for passive flow stabilization over broad frequency ranges. Standard formulations for the stability problem in the presence of compliant surfaces often rely on the Orr-Sommerfeld equation coupled with a pre-defined wall admittance. However, this decoupling typically restricts the problem to a spatial stability analysis where the frequency must be fixed a priori, often neglecting the complex interplay between fluid and solid modes. By employing a monolithic variational approach, we naturally enforce the continuity of velocity and traction forces at the fluid-solid interface without the need for ad-hoc interface conditions or impedance functions. The resulting discretized system yields a coupled generalized eigenvalue problem. A key advantage of this formulation is its versatility: it allows for the determination of complex phase velocities and growth rates in both spatial frameworks (imposed frequency) and temporal frameworks (imposed wavenumber). Furthermore, it fully accommodates the geometric complexity required to model intricate metamaterial designs, which can be both frequency and wavenumber-dependent. Our numerical analysis focuses on the energy exchange between fluid instabilities and elastic waves in the metamaterial structure. Specifically, we examine how the phononic band structure influences the synchronization between TS waves in the fluid and traveling waves in the solid. Preliminary results indicate that the fully coupled temporal stability analysis yields distinct outcomes compared to uncoupled spatial analysis, even for identical metamaterial designs. This discrepancy suggests that full coupling is essential for accurate prediction, opening new avenues for passive flow control and drag reduction.