  The quantum Grothendieck ring of a certain category of finite-dimensional
modules over a quantum loop algebra associated with a complex
finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster
algebra structure of skew-symmetric type. Partly motivated by a search of a
ring corresponding to a quantum cluster algebra of {\em skew-symmetrizable}
type, the quantum {\em virtual} Grothendieck ring, denoted by
$\mathfrak{K}_q(\mathfrak{g})$, is recently introduced by Kashiwara--Oh
\cite{KO23} as a subring of the quantum torus based on the $(q,t)$-Cartan
matrix specialized at $q=1$. In this paper, we prove that
$\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structure
of skew-symmetrizable type. This task essentially involves constructing
distinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to make
cluster variables and generalizing the quantum $T$-system associated with
Kirillov--Reshetikhin modules to establish a quantum exchange relation of
cluster variables. Furthermore, these distinguished bases naturally fit into
the paradigm of Kazhdan--Lusztig theory and our study of these bases leads to
some conjectures on quantum positivity and $q$-commutativity.
