  We obtain the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and
anti-BRST symmetry transformations (corresponding to the infinitesimal
classical gauge symmetry transformations) for the St${\ddot
u}$ckelberg-modified massive three $(2 + 1)$-dimensional (3D) Abelian 2-form
gauge theory with a single pseudo-scalar field. The latter field (having the
negative kinetic term and a well-defined mass) has already been shown (i) to
exist in the modified version of the standard 3D St${\ddot u}$ckelberg
formalism (on solid mathematical grounds), (ii) to be a possible candidate for
dark matter, and (iii) to correspond to the ``phantom" field of some of the
cosmological models of the Universe. A couple of highlights of our present
endeavor are (i) the observation that, even though the pseudo-scalar field does
not transform under the gauge and (anti-)BRST symmetry transformations, it
appears in the first-class constraints which annihilate the physical states at
the quantum level, and (ii) the Noether conserved (anti-)BRST charges are found
to be non-nilpotent. In our present investigation, we derive (i) the coupled
(but equivalent) BRST and anti-BRST invariant Lagrangian densities, (ii) the
conserved and off-shell nilpotent versions of the (anti-)BRST charges and the
conserved ghost charge, (iii) the (anti-)BRST invariant Curci-Ferrari (CF) type
restrictions, and (iv) the standard BRST algebra amongst the conserved and
nilpotent (anti-)BRST and conserved ghost charges of our theory.
