  We present the photometric and spectroscopic evolution of SN 2022oqm, a
nearby multi-peaked hydrogen- and helium-weak calcium-rich transient (CaRT). SN
2022oqm was detected 13.1 kpc from its host galaxy, the face-on spiral galaxy
NGC 5875. Extensive spectroscopic coverage reveals an early hot (T >= 40,000 K)
continuum and carbon features observed $\sim$1~day after discovery, SN Ic-like
photospheric-phase spectra, and strong forbidden calcium emission starting 38
days after discovery. SN 2022oqm has a relatively high peak luminosity (MB =
-17 mag) for (CaRTs), making it an outlier in the population. We determine that
three power sources are necessary to explain the light curve (LC), with each
corresponding to a distinct peak. The first peak is powered by an expanding
blackbody with a power law luminosity, suggesting shock cooling by
circumstellar material (CSM). Subsequent LC evolution is powered by a double
radioactive decay model, consistent with two sources of photons diffusing
through optically thick ejecta. From the LC, we derive an ejecta mass and 56Ni
mass of ~0.6 solar masses and ~0.09 solar masses. Spectroscopic modeling
suggests 0.6 solar masses of ejecta, and with well-mixed Fe-peak elements
throughout. We discuss several physical origins for SN 2022oqm and find either
a surprisingly massive white dwarf progenitor or a peculiar stripped envelope
model could explain SN 2022oqm. A stripped envelope explosion inside a dense,
hydrogen- and helium-poor CSM, akin to SNe Icn, but with a large 56Ni mass and
small CSM mass could explain SN 2022oqm. Alternatively, helium detonation on an
unexpectedly massive white dwarf could also explain SN 2022oqm.
