A nonvanishing entropy production rate is one of the defining characteristics of any nonequilibrium system, and several techniques exist to determine this quantity directly from experimental data. The short-time inference scheme, derived from the thermodynamic uncertainty relation, is a recent addition to the list of these techniques. Here we apply this scheme to quantify the entropy production rate in a class of microscopic heat engine models called Brownian gyrators. In particular, we consider models with anharmonic confining potentials. In these cases, the dynamical equations are indelibly nonlinear, and the exact dependencies of the entropy production rate on the model parameters are unknown. Our results demonstrate that the short-time inference scheme can efficiently determine these dependencies from a moderate amount of trajectory data. Furthermore, the results show that the nonequilibrium properties of the gyrator model with anharmonic confining potentials are considerably different from its harmonic counterpart; especially in setups leading to a nonequilibrium dynamics and the resulting gyration patterns.