The synthetic construction of intracellular circuits is frequently hindered by a poor knowledge of appropriate kinetics and precise rate parameters. Here, we use generalized modeling (GM) to study the dynamical behavior of topological models of a family of hybrid metabolic-genetic circuits known as "metabolators." Under mild assumptions on the kinetics, we use GM to analytically prove that all explicit kinetic models which are topologically analogous to one such circuit, the "core metabolator," cannot undergo Hopf bifurcations. Then, we examine more detailed models of the metabolator. Inspired by the experimental observation of a Hopf bifurcation in a synthetically constructed circuit related to the core metabolator, we apply GM to identify the critical components of the synthetically constructed metabolator which must be reintroduced in order to recover the Hopf bifurcation. Next, we study the dynamics of a re-wired version of the core metabolator, dubbed the "reverse" metabolator, and show that it exhibits a substantially richer set of dynamical behaviors, including both local and global oscillations. Prompted by the observation of relaxation oscillations in the reverse metabolator, we study the role that a separation of genetic and metabolic time scales may play in its dynamics, and find that widely separated time scales promote stability in the circuit. Our results illustrate a generic pipeline for vetting the potential success of a circuit design, simply by studying the dynamics of the corresponding generalized model. The engineering of biological circuits, synthetically constructed in the laboratory and designed to exhibit novel behaviors, has matured rapidly as a discipline over the past decade. A major challenge for synthetic biology is understanding how different cellular subsystems, composed of distinct components and operating at widely different speeds, interface and work together to generate coherent cellular behaviors. In particular, metabolic pathways can rapidly harvest energy and nutrients for reproduction, while genetic regulatory networks slowly control these pathways in response to environmental changes. Here, we study a generalized version of the metabolator, a unique synthetic circuit composed of both metabolic and regulatory components, previously constructed experimentally and shown to exhibit oscillations. In addition to exploring multiple alternative network topologies, our generalized approach overcomes the inherent uncertainty associated with precise kinetic details of biological circuits. Through this analysis, we identify circuit components that are essential for producing sustained oscillations. Furthermore, we find that certain dynamical regimes of a rewired metabolator display unexpected oscillatory properties, such as the capacity to suddenly transition to high amplitude oscillations. This novel behavior could be tested experimentally, and lead to novel synthetic biology modules and applications.