The question of characterization of the degree of nonequilibrium activity in active matter systems is studied in the context of a stochastic microswimmer model driven by a chemical cycle. The resulting dynamical properties and entropy production rate unravel a complex interplay between the chemical and the hydrodynamic degrees of freedom beyond linear response, which is not captured by conventional phenomenological approaches. By studying the precision-dissipation trade off, a new protocol is proposed in which microscopic chemical driving forces can be inferred experimentally. Our findings highlight subtleties associated with the stochastic thermodynamics of autonomous microswimmers. Published by the American Physical Society 2024