  The 2017 Event Horizon Telescope (EHT) observations of M87* detected a
ring-shaped feature $\sim40\mu$as in diameter, consistent with the event
horizon scale of a black hole of the expected mass. The thickness of this ring,
however, proved difficult to measure, despite being an important parameter for
constraining the observational appearance. In the first paper of this series we
asked whether the width of the ring was sensitive to the choice of likelihood
function used to compare observed closure phases and closure amplitudes to
model predictions. In this paper we investigate whether the ring width is
robust to changes in the model itself. We construct a more realistic geometric
model with two new features: an adjustable radial falloff in brightness, and a
secondary "photon ring" component in addition to the primary annulus. This
thin, secondary ring is predicted by gravitational lensing for any black hole
with an optically thin accretion flow. Analyzing the data using the new model,
we find that the primary annulus remains narrow (fractional width $\leq0.25$)
even with the added model freedom. This provides further evidence in favor of a
narrow ring for the true sky appearance of M87*, a surprising feature that, if
confirmed, would demand theoretical explanation. Comparing the Bayesian
evidence for models with and without a secondary ring, we find no evidence for
the presence of a lensed photon ring in the 2017 observations. However, the
techniques we introduce may prove useful for future observations with a larger
and more sensitive array.
