The desired integral is given in equation (13) of <A HREF="https://arxiv.org/abs/1812.06069">arXiv:1812.06069</A>:

$$\int_{{SU}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\sum_{q=-\infty}^\infty\prod_{i=1}^N\delta_{\lambda_i,\mu_i+q}.$$