  Operating the cavity-embedded Cooper pair transistor (cCPT) in the Kerr
bistable regime, we demonstrate single-shot resolution between two charge
states that are $0.09e$ apart. The measurement is performed with 94$\%$
fidelity in a duration of 3 $\mu$s. The drive power at which the measurement is
performed corresponds to only 20 intracavity photons on average in the high
oscillation amplitude state of the cCPT, which is orders-of-magnitude smaller
than that in rf-SETs. We find that the limiting factor for this mode of
operation of the cCPT is the spontaneous fluctuation-induced switching between
the two metastable oscillation amplitude states. We present empirical data on
the variation of the switching dynamics with drive parameters and cCPT DC bias.
