Gauge-invariant descriptions for a free bosonic scalar field of continuous spin in a $d$-dimensional Minkowski space-time using a metric-like formulation are constructed on the basis of a constrained BRST-BFV approach we propose. The resulting BRST-BFV equations of motion for a scalar field augmented by ghost operators contains different sets of auxiliary fields, depending on the manner of a partial gauge-fixing and a resolution of some of the equations of motion for a BRST-unfolded first-stage reducible gauge theory. To achieve an equivalence of the resulting BRST-unfolded constrained equations of motion with the initial irreducible Poincare group conditions of a Bargmann--Wigner type, it is demonstrated that one should replace the field in these conditions by a class of gauge-equivalent configurations. Triplet-like, doublet-like constrained descriptions, as well as an unconstrained quartet-like non-Lagrangian and Lagrangian formulations, are derived using both Fronsdal-like and new tensor fields. In particular, the BRST--BV equations of motion and Lagrangian using an appropriate set of Lagrangian multipliers in the minimal sector of the respective field and antifield configurations are constructed in a manifest way.
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