  The leading explanation of the $\textit{Fermi}$ Galactic center $\gamma$-ray
excess is the extended emission from a unresolved population of millisecond
pulsars (MSPs) in the Galactic bulge. Such a population would, along with the
prompt $\gamma$ rays, also inject large quantities of electrons/positrons
($e^\pm$) into the interstellar medium. These $e^\pm$ could potentially
inverse-Compton (IC) scatter ambient photons into $\gamma$ rays that fall
within the sensitivity range of the upcoming Cherenkov Telescope Array (CTA).
In this article, we examine the detection potential of CTA to this signature by
making a realistic estimation of the systematic uncertainties on the Galactic
diffuse emission model at TeV-scale $\gamma$-ray energies. We forecast that, in
the event that $e^\pm$ injection spectra are harder than $E^{-2}$, CTA has the
potential to robustly discover the IC signature of a putative Galactic bulge
MSP population sufficient to explain the GCE for $e^\pm$ injection efficiencies
in the range $\approx 2.9-74.1\%$, or higher, depending on the level of
mismodeling of the Galactic diffuse emission components. On the other hand, for
spectra softer than $E^{-2.5}$, a reliable CTA detection would require an
unphysically large $e^\pm$ injection efficiency of $\gtrsim 158\%$. However,
even this pessimistic conclusion may be avoided in the plausible event that MSP
observational and/or modeling uncertainties can be reduced. We further find
that, in the event that an IC signal were detected, CTA can successfully
discriminate between an MSP and a dark matter origin for the radiating $e^\pm$.
