  Within the conformally reduced gravity model, where the metric is
parametrised by a function $f(\phi)$ of the conformal factor $\phi$, we keep
dependence on both the background and fluctuation fields, to local potential
approximation and $\mathcal{O}(\partial^2)$ respectively, making no other
approximation. Explicit appearances of the background metric are then dictated
by realising a remnant diffeomorphism invariance. The standard non-perturbative
Renormalization Group (RG) scale $k$ is inherently background dependent, which
we show in general forbids the existence of RG fixed points with respect to
$k$. By utilising transformations that follow from combining the flow equations
with the modified split Ward identity, we uncover a unique background
independent notion of RG scale, $\hat k$. The corresponding RG flow equations
are then not only explicitly background independent along the entire RG flow
but also explicitly independent of the form of $f$. In general $f(\phi)$ is
forced to be scale dependent and needs to be renormalised, but if this is
avoided then $k$-fixed points are allowed and furthermore they coincide with
$\hat k$-fixed points.
