  Gravitational lensing offers a competitive method to measure $H_0$ with the
goal of 1% precision. A major obstacle comes in the form of lensing
degeneracies, such as the mass sheet degeneracy (MSD), which make it possible
for a family of density profiles to reproduce the same lensing observables but
return different values of $H_0$. The modeling process artificially selects one
choice from this family, potentially biasing $H_0$. The effect is more
pronounced when the profile of a given lens is not perfectly described by the
lens model, which will always be the case to some extent. To explore this, we
quantify the bias and spread in $H_0$ by creating quads from two-component mass
models and fitting them with a power-law ellipse+shear model. We find that the
bias does not correspond to the estimate one would calculate by transforming
the profile into a power law near the image radius. We also emulate the effect
of including stellar kinematics by performing fits where the slope is
constrained to the true value. Informing the fit using the true value near the
image radius can introduce substantial bias (0-23% depending on the model). We
confirm using Jeans arguments that kinematic constraints can result in a biased
value of $H_0$, though the degree of bias depends on the region kinematic
modeling probes in specific lenses. We conclude that lensing degeneracies
manifest through commonplace modeling approaches in a more complicated way than
is assumed in the literature. If stellar kinematics incorrectly break the MSD,
their inclusion may introduce more bias than their omission.
