Alternating (AC) magnetic fields can induce not only an alternating magnetization in materials, but also an alternating temperature via the magnetocaloric effect. The latter effect is typically neglected when performing AC susceptibility measurements, but consideration of both effects on an equal footing is necessary in order to reliably distinguish between internal and external causes of magnetic response and accurately extract quantitative information about relaxation processes. In order to address this, we have developed a method to measure the AC magnetocaloric effect that is compatible with AC susceptibility measurements, and also a framework to analyze these data in combination. We demonstrate the efficacy of this approach using YbVO4, a material for which strong single-ion anisotropy leads to slow electronic spin-lattice relaxation at low temperatures via a phonon bottleneck effect. We report AC magnetic susceptibility and AC magnetocaloric effect measurements for this material as a function of field and frequency at a temperature of 3 K. We analyze the data using a discretized thermal model, and extract the field-dependence of the intrinsic spin-lattice relaxation rate. This demonstration experiment illustrates a general approach to quantitatively address multiple measured quantities in driven systems using a unified thermal circuit analysis. The thermal analysis methods presented in this report can be extended to study other magnetic, dielectric, and elastic materials exhibiting a complex response to an external driving field in the presence of internal and external relaxation, particularly when an energy dissipation process is within an accessible frequency regime.