Sensing using surface plasmon resonances is one of the most promising practical applications of plasmonic nanostructures and Fano resonances allow achieving a lower detection limit thanks to their narrow spectral features. However, a narrow spectral width of the subradiant mode in a plasmonic system, as observed in the weak coupling regime, is in general associated with a low modulation of the complete spectral response. In this article, we show that this limitation can be overcome by a nonlinear approach based on second harmonic generation and its dependence on symmetry at the nanoscale. The Fano resonant systems considered in this work are gold nanodolmens. Their linear and nonlinear responses are evaluated using a surface integral equation method. The numerical results demonstrate that a variation of the refractive index of the surrounding medium modifies the coupling between the dark and bright modes, resulting in a modification of the electromagnetic wave scattered at the second harmonic wavelength, especially the symmetry of the nonlinear emission. Reciprocally, we show that evaluating the asymmetry of the nonlinear emission provides a direct measurement of the gold nanodolmens dielectric environment. Interestingly, the influence of the refractive index of the surrounding medium on the nonlinear asymmetry parameter is approximately 10 times stronger than on the spectral position of the surface plasmon resonance: hence, smaller refractive index changes can be detected with this new approach. Practical details for an experimental realization of this sensing scheme are discussed and the resolution is estimated to be as low as Delta n = 1.5 x 10(-3), respectively 1.5 x 10(-5), for an acquisition time of 60 s for an isolated gold nanodolmen, respectively an array of 10 x 10 nanodolmens.