We develop a first-principles model of the regenerator component of a generic Stirling engine. The model is based on the Euler equations of one-dimensional gas dynamics coupled with its convective/conductive heat transfer with the embedded mesh material. We investigate various methods for deriving simpler and low-order control-oriented models from this first principles model, the basic criterion being high fidelity representation of the dynamics of the regenerator when coupled to other dynamic components of the engine. We identify several nondimensional parameters that potentially categorize different modes of operation, and investigate the corresponding time-scale separation. A hierarchy of singularly perturbed models is derived in which acoustic dynamics are eliminated, periodic mesh dynamics are averaged, and the shape of the distributed regenerator gas state is approximated. In addition, since the reduced model is to be operated cyclically when connected to other parts of the engine, we develop such a feedback-aware model reduction algorithm based on a proper orthogonal decomposition (POD) with a chirped signal input (chirp-POD). This algorithm yields reduced models that are accurate over a range of engine operating frequencies.