Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_n\})$ of the unitary symplectic matrix $U$, via the metric tensor,
$$g_{mn}=-{\rm tr}\,U^{\dagger}(\partial U/\partial\alpha_m)U^{\dagger}(\partial U/\partial\alpha_n),$$
and then the Haar measure is $d\mu= \sqrt{{\rm det}\,g}\prod_n d\alpha_n$.
Alternatively, you can use a computer to generate random matrices with the desired measure, as explained by Franco Mezzadri.