Bayesian analyses in the context of relativistic heavy-ion collisions have so far relied almost exclusively on bulk hadronic observables constructed from momentum degrees of freedom to constrain the transport properties of the quark-gluon plasma. In this work, we perform the Bayesian inference after incorporating the longitudinal spin polarization of $\Lambda$ hyperons alongside conventional bulk measurements in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV to constrain the shear and bulk viscosity to entropy density ratios, $\eta/s$ and $\zeta/s$. We demonstrate that the inclusion of spin polarization, which provides complementary sensitivity to the space-time structure and vorticity of the medium, shifts the posterior distribution of $\zeta/s$ toward larger values, although current uncertainties do not allow a statistically significant separation at the 68% credibility level. Nevertheless, the results establish spin polarization as a valuable probe in quantitative studies of QGP transport properties and indicate that it should be incorporated in comprehensive and systematically constrained Bayesian extractions of the medium's dynamical parameters.