is mixed between two pairs of quadratures of a bosonic mode and a spin. In this setting, we can suppress the dispersive shift while retaining a nonzero Kerr shift, which remarkably results in a cubic relationship between shot noise dephasing and thermal photons in the oscillator. We demonstrate this configuration with a simple toy model, quantify the expected improvements to photon shot-noise dephasing of the spin, and describe an approach to fast qubit readout via the Kerr shift. Further, we show how such a regime is achievable in superconducting circuits because magnetic and electric couplings can be of comparable strength, using two examples: the Cooper pair transistor and the fluxonium molecule.