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Operating the cavity-embedded Cooper pair transistor (cCPT) in the Kerrbistable regime, we demonstrate single-shot resolution between two chargestates 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 isperformed corresponds to only 20 intracavity photons on average in the highoscillation amplitude state of the cCPT, which is orders-of-magnitude smallerthan that in rf-SETs. We find that the limiting factor for this mode ofoperation of the cCPT is the spontaneous fluctuation-induced switching betweenthe two metastable oscillation amplitude states. We present empirical data onthe variation of the switching dynamics with drive parameters and cCPT DC bias.

Fast high-fidelity charge readout by operating the cavity-embedded Cooper pair transistor in the Kerr bistable regime