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Recent theoretical studies showed that the electronic structure of 1$T$-TaS$_2$ in the low-temperature commensurate charge density wave phase exhibits a nontrivial interplay between band-insulating and Mott insulating behavior. This has important implications for the interpretation of photo-doping experiments. Here we use nonequilibrium dynamical mean-field theory simulations of a realistic multi-layer structure to clarify the charge carrier dynamics induced by a laser pulse. The solution is propagated up to the picosecond timescale by employing a memory-truncation scheme. While long-lived doublons and holons only exist in the surface state of a specific structure, the disturbance of bonding states in the bilayers which make up the bulk of the system explain the almost instantaneous appearance of in-gap states. Our simulations consistently explain the coexistence of a doublon feature with a prominent ``background" signal in previous time-resolved photoemission experiments, and they suggest strategies for the selective population of the ingap and doublon states by exploiting the sensitivity to the pump polarization and pump frequency.