When studying the spin-noise-induced fluctuations of Kerr rotation in a quantum-well microcavity, we have found a dramatic increase of the noise signal (by more than two orders of magnitude) in the vicinity of anti-crossing of the polariton branches. The effect is explained by nonlinear optical instability of the microcavity giving rise to the light-power-controlled amplification of the polarization noise signal. In the framework of the developed model of built-in amplifier, we also interpret the nontrivial spectral and intensity-related properties of the observed noise signal below the region of anti-crossing of polariton branches. The discovered effect of optically controllable amplification of broadband polarization signals in microcavities in the regime of optical instability may be of interest for detecting weak oscillations of optical anisotropy in fundamental research and for other applications in optical information processing.