  Feedback by active galactic nuclei (AGNs) is essential for regulating the
fast radiative cooling of low-entropy gas at the centers of galaxy clusters and
for reducing star formation rates of central ellipticals. The details of
self-regulation depend critically on the unknown contents of AGN-inflated
bubbles. Observations of the Sunyaev-Zeldovich (SZ) signal of AGN bubbles
provide us with the ability to directly measure the lobe electron pressure
given a bubble morphology. Here we compute the SZ signal of jet-inflated
bubbles in three-dimensional magnetohydrodynamical simulations of the galaxy
cluster MS0735.6+7421 with the Arepo code, and compare our synthetic SZ results
to inferences obtained with popular modelling approaches. We find that cutting
out ellipsoidal bubbles from a double-beta pressure profile only matches the
inner bubble edges in the simulations and fails to account for the emission of
the shock-enhanced pressure cocoon outside the bubbles. This additional
contribution significantly worsens the accuracy of the cut-out method for jets
with small inclinations with respect to the line of sight. Also, the kinetic SZ
effect of the bubbles, a previously neglected contribution, becomes relevant at
these smaller inclinations due to entrainment and mixing of the intracluster
medium with low-density jet material. Fortunately, the different signs of the
kinetic SZ signal in opposite lobes allow modelling this effect. We present an
approximate method to determine the jet inclination, which combines jet power
and lifetime estimates, the stand-off distance between jet head and bow shock,
and the kinetic SZ effect, thereby helping to correctly infer the bubble
contents.
