We consider fluid flows, governed by the Navier-Stokes equations, subject to a steady symmetry-breaking bifurcation and forced by a weak noise acting on a slow timescale. By generalizing the multiple-scale weakly nonlinear expansion technique employed in the literature for the response of the Duffing oscillator, we rigorously derive a stochastically forced Stuart-Landau equation for the dominant symmetry-breaking mode. The probability density function of the solution, and of the escape time from one attractor to the other, are then determined by solving the associated Fokker-Planck equation. The validity of this reduced order model is tested on the flow past a sudden expansion for a given Reynolds number and different noise amplitudes. At a very low numerical cost, the statistics obtained from the amplitude equation accurately reproduce those of long-time direct numerical simulations.