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

$$\int_{{SU}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\sum_{q=-\infty}^\infty\prod_{i=1}^N\delta_{\lambda_i,\mu_i+q},$$
where $\lambda=(\lambda_1,\lambda_2,\ldots\lambda_N)$ and $|\lambda|=\sum_{i}\lambda_i$, with $\lambda_1\geq\lambda_2\cdots\geq 0$.

The $SU(N)$ integral vanishes unless $|\lambda|=|\mu|$ modulo $N$. For $|\lambda|=|\mu|$ the $U(N)$ and $SU(N)$ integrals are identical.