Observations with the JWST in lensed fields have revealed that galaxies at cosmic dawn may concentrate their star formation in highly dense, compact, star clusters. The high columns and low metallicities encountered in their birth environments suggest that Lyman-alpha (Lyα) radiation pressure may be crucial to their formation and evolution. In this study, we address this question by post-processing snapshots from radiation hydrodynamic simulations of dense star cluster-forming clouds (Σ_*≳10³M_⊙pc⁻²) with a range of dust abundances (Z_d=0-0.1Z_(d,⊙)) using the COLT Monte Carlo code. We infer that Lyα is likely to have mild ( 10%) effects on the gas-to-star conversion efficiencies (ε_*≳60%) for Z_d≳0.01Z_(d,⊙), and even in dust-free environments, ε_*≳25% - much higher than the <10% values typical of star-forming regions in the local Universe. This is because the densest filaments dominating stellar mass assembly (n≳10⁴cm⁻³) remain sub-Eddington (f_Edd<1). On the other hand, the bulk of the gas volume (n≲10³cm⁻³) has f_Edd>1, with noticeable fractions having f_Edd≳10, implying that Lyα can launch dynamically significant winds from these systems rapidly (≲4Myr), with possible implications for ionizing photon escape and galactic outflows. The Lyα force multiplier M_F is highly sensitive to Z_d, with M_F≲3 (≲ 500) for 0.1Z_(d,⊙) (dust-free) environments respectively. Nevertheless, Lyα dominates over UV and IR radiation pressure at all values of Z_d≲0.1Z_(d,⊙), by factors of ~3-500. Our results suggest that Lyα radiation pressure reinforces the emerging picture of locally efficient, bursty star formation accompanied by rapid outflows in galaxies at cosmic dawn.