  We consider discretisations of the Macdonald--Mehta integrals from the theory
of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$,
$\mathrm{B}_r$ and $\mathrm{D}_r$, we provide closed-form evaluations in those
cases for which the Weyl denominators featuring in the summands have exponents
$1$ and $2$. Our proofs for the exponent-$1$ cases rely on identities for
classical group characters, while most of the formulas for the exponent-$2$
cases are derived from a transformation formula for elliptic hypergeometric
series for the root system $\mathrm{BC}_r$. As a byproduct of our results, we
obtain closed-form product formulas for the (ordinary and signed) enumeration
of orthogonal and symplectic tableaux contained in a box.
