  In formulating a non-orientable analogue of the Milnor Conjecture on the
$4$-genus of torus knots, Batson developed an elegant construction that
produces a smooth non-orientable spanning surface in $B^4$ for a given torus
knot in $S^3$. While Lobb showed that Batson's surfaces do not always minimize
the non-orientable $4$-genus, we prove that they always do minimize among
surfaces that share their normal Euler number. We also completely determine the
possible pairs of normal Euler number and first Betti number for non-orientable
surfaces whose boundary lies in a class of torus knots for which Batson's
surfaces are non-orientable $4$-genus minimizers.
