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 $\sum_\mu m_\mu\mu$ (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 we have $\det \circ R = \det(\cdot)^n$.

Inspired by [your comment](https://mathoverflow.net/questions/439350/sum-of-weights-of-an-irreducible-representation-of-un#comment1133302_439350), I realise we can be a little more explicit.  (On having written this, I realise that it was actually *exactly* what you were saying; I mistook your $\lambda_i$ for an indexing of weights, rather than a component of $\lambda$.  Oops, sorry!)

Let $\lambda$ be the highest weight of $R$.  Then all weights $\mu$ of $R$ agree on the centre of $\operatorname U(N)$ with $\lambda$.  Specifically, they all act as $z I_N \mapsto z^\ell$ for some integer $\ell$.  (If we think of $\lambda$ as an element of $\mathbb Z^N$, then $\ell$ is the sum of the components.)  Thus, since $\sum_\mu m_\mu$ equals $\dim(R)$, we have that $R$ agrees on the centre with $z I_N \mapsto z^{\ell\dim(R)}I_{\dim(R)}$, so $\det \circ R$ agrees on the centre with $z I_N \mapsto z^{N\ell\dim(R)} = \det(z I_N)^{\ell\dim(R)}$, so $A(R)$ equals $\ell\dim(R)$.  As you [point out](https://mathoverflow.net/questions/439350/sum-of-weights-of-an-irreducible-representation-of-un#comment1133302_439350), we can use the [Weyl dimension formula](https://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_dimension_formula) to compute $\dim(R)$ in terms of $\lambda$ if desired.