  A constrained BRST-BV Lagrangian formulation for totally symmetric massless
HS fields in a $d$-dimensional Minkowski space is extended to a non-minimal
constrained BRST-BV Lagrangian formulation by using a non-minimal BRST operator
$Q_{c|\mathrm{tot}}$ with non-minimal Hamiltonian BFV oscillators
$\overline{C}, \overline{\mathcal{P}}, \lambda, \pi$, as well as antighost and
Nakanishi-Lautrup tensor fields, in order to introduce an admissible
self-consistent gauge condition. The gauge-fixing procedure involves an
operator gauge-fixing BRST-BFV Fermion $\Psi_H$ as a kernel of the gauge-fixing
BRST-BV Fermion functional $\Psi$, manifesting the concept of BFV-BV duality. A
Fock-space quantum action with non-minimal BRST-extended off-shell constraints
is constructed as a shift of the total generalized field-antifield vector by a
variational derivative of the gauge-fixing Fermion $\Psi$ in a total BRST-BV
action $S^{\Psi}_{0|s} = \int d \eta_0 \langle \chi^{\Psi{} 0}_{\mathrm{tot}|c}
\big| Q_{c|\mathrm{tot}}\big| \chi^{\Psi{} 0}_{\mathrm{tot}|c}\rangle$. We use
a gauge condition which depends on two gauge parameters, thereby extending the
case of $R_\xi$-gauges. For triplet and duplet formulations we explored the
representations with only traceless field-antifield and source variables. For
the generating functionals of Green's functions, BRST symmetry transformations
are suggested and Ward identities are obtained.
