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Recently, Co-based honeycomb magnets have been proposed as promising candidate materials to host the Kitaev spin liquid state. One of the front-runners is BaCo₂(AsO₄)₂ (BCAO), where it was suggested that the exchange processes between Co²⁺ ions via the surrounding edge-sharing oxygen octahedra could give rise to bond-dependent Kitaev interactions. In this work, we present and analyze comprehensive inelastic neutron scattering studies of BCAO with fields in the honeycomb plane. Combining the constraints from the magnon excitations in the high-field polarized state and the inelastic spin structure factor measured in zero magnetic field, we examine two leading theoretical models: the Kitaev-type JKΓΓ' model and the XXZ-J₁J₃ model. We show that the existing experimental data can be consistently accounted for by the XXZ-J₁J₃ model but not by the JKΓΓ' model, and we discuss the implications of these results for the realization of a spin liquid phase in BCAO and more generally for the realization of the Kitaev model in cobaltates.