  The quantum phase transition to a $\mathbb{Z}_3$-ordered Kekul\'e valence
bond solid in two-dimensional Dirac semimetals is governed by a fermion-induced
quantum critical point, which renders the putatively discontinuous transition
continuous. We study the resulting universal critical behavior in terms of a
functional RG approach, which gives access to the scaling behavior on the
symmetry-broken side of the phase transition, for general dimension and number
of Dirac fermions. In particular, we investigate the emergence of the
fermion-induced quantum critical point for space-time dimensions $2<d<4$. We
determine the integrated RG flow from the Dirac semi-metal to the
symmetry-broken regime and analyze the underlying fixed point structure. We
show that the fermion-induced criticality leads to a scaling form with two
divergent length scales, due to the breaking of the discrete $\mathbb{Z}_3$
symmetry. This provides another source of scaling corrections, besides the one
stemming from being in the proximity to the first order transition.
