As discussed in the [comments](https://mathoverflow.net/questions/439350/sum-of-weights-of-an-irreducible-representation-of-un#comment1133244_439350), your sum is a Weyl-fixed character, so trivial for $G = \operatorname{SU}(N)$ and a multiple of $\det = (1, \dotsc, 1)$ for $\operatorname U(N)$.

To be concrete, as I guessed in the [comments](https://mathoverflow.net/questions/439350/sum-of-weights-of-an-irreducible-representation-of-un#comment1133257_439350), one sees that your sum $\sum m_\mu\mu$ (the sum taking place in the character lattice $X^*(T)$ of the (implicitly chosen) maximal torus $T$, not in $\mathbb C[X^*(T)]$ as in the Weyl character formula) is precisely the character of $\det \circ R$.  Again, since $\det \circ R$ is Weyl invariant and $\det$ spans the Weyl-invariant part of the character lattice, your integer $A(R)$ is precisely the integer $N$ such that $\det \circ R = \det(\cdot)^N$.