  We introduce the cavity-embedded Cooper pair transistor (cCPT), a device
which behaves as a highly nonlinear microwave cavity whose resonant frequency
can be tuned both by charging a gate capacitor and by threading flux through a
SQUID loop. We characterize this device and find excellent agreement between
theory and experiment. A key difficulty in this characterization is the
presence of frequency fluctuations comparable in scale to the cavity linewidth,
which deform our measured resonance circles in accordance with recent
theoretical predictions [Brock et al., Phys. Rev. Applied 14, 054026 (2020)].
By measuring the power spectral density of these frequency fluctuations at
carefully chosen points in parameter space, we find that they are primarily a
result of the $1/f$ charge and flux noise common in solid state devices.
Notably, we also observe key signatures of frequency fluctuations induced by
quantum fluctuations in the cavity field via the Kerr nonlinearity.
