The leading explanation of the Fermi Galactic center 𝛾-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 𝛾 rays, also inject large quantities of electrons/positrons (𝑒 ± ) into the interstellar medium. These 𝑒 ± could potentially inverse-Compton (IC) scatter ambient photons into 𝛾 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 𝛾-ray energies. We forecast that, in the event that 𝑒 ± 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 ± injection efficiencies in the range ≈ 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 ± injection efficiency of 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 ± .