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 Greens functions, BRST symmetry transformations are suggested and Ward identities are obtained.
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