  We introduce sets $\mathcal{R}(\sigma)$ consisting of semistandard tableaux
of shape $\sigma$, subject to a certain restriction on the initial column.
These $\mathcal{R}(\sigma)$ are generalizations of the rational, symplectic, or
orthogonal tableaux (which furnish a weight basis for the irreducible
finite-dimensional representations $U_\sigma$ of the classical groups $H =
\operatorname{GL}_k$, $\operatorname{Sp}_{2k}$, or $\operatorname{O}_k$,
respectively). As our main result, we show that the cardinality of
$\mathcal{R}(\sigma)$ gives the Bernstein degree (i.e., multiplicity) of the
modules of covariants (of type $U_\sigma$) of each classical group $H$. Via
Howe duality, these modules can also be viewed as $(\mathfrak{g},K)$-modules of
unitary highest weight representations of a real reductive group
$G_{\mathbb{R}}$. Notably, our result using $\mathcal{R}(\sigma)$ is valid
regardless of the rank of $H$, whereas the previous result of
Nishiyama-Ochiai-Taniguchi (expressing the Bernstein degree in terms of $\dim
U_\sigma$) holds only when $k$ is at most the real rank of $G_{\mathbb{R}}$. In
effect, as $k$ increases beyond this range, $|\mathcal{R}(\sigma)|$
interpolates between $\dim U_\sigma$ and the dimension of a certain "limiting"
$K$-module.
