  We numerically and analytically investigate the behavior of a non-equilibrium
phase transition in the second Schl\"ogl autocatalytic reaction scheme. Our
model incorporates both an interaction-induced phase separation and a
bifurcation in the reaction kinetics, with these critical lines coalescing at a
bicritical point in the macroscopic limit. We construct a stochastic master
equation for the reaction processes to account for the presence of mutual
particle interactions in a thermodynamically consistent manner by imposing a
generalized detailed balance condition, which leads to exponential corrections
for the transition rates. In a non-spatially extended (zero-dimensional)
setting, we treat the interactions in a mean-field approximation, and introduce
a minimal model that encodes the physical behavior of the bicritical point and
permits the exact evaluation of the anomalous scaling for the particle number
fluctuations in the thermodynamic limit. We obtain that the system size scaling
exponent for the particle number variance changes from $\beta_0 = 3/2$ at the
standard non-interacting bifurcation to $\beta = 12/7$ at the interacting
bicritical point. The methodology developed here provides a generic route for
the quantitative analysis of fluctuation effects in chemical reactions
occurring in multi-component interacting systems.
