  We investigate the $\alpha$-$\beta$ mapping, as previously introduced by
Islam et al.~\cite{Islam:2022laz}, which relates numerical relativity (NR) and
adiabatic point-particle black hole perturbation theory (BHPT) waveforms in the
comparable mass regime for quasi-circular, non-spinning binary black holes.
This mapping involves scaling the amplitude of individual modes with different
values of $\alpha$ and the time (and therefore the phase) with a single
parameter, $\beta$. In this paper, we demonstrate that this scaling, both in
terms of time and orbital frequencies, also extends to the overall
gravitational-wave energy flux. This means that we can find a single
$\alpha_{\mathcal{F}}$ that scales the BHPT flux and a single
$\beta_{\mathcal{F}}$ (which matches the value of $\beta$) that scales the BHPT
time such a way that it aligns with NR flux evolution. We then explore the
connection between the scaling parameter $\alpha_{\mathcal{F}}$
($\beta_{\mathcal{F}}$) and the missing finite size correction for the
secondary black hole within the BHPT framework.
