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X-rays from High-Mass X-ray Binaries (HMXBs) are likely the main source of heating of the intergalactic medium (IGM) during Cosmic Dawn (CD), before the completion of reionization. This Epoch of Heating (EoH; $z\sim 10-15$) should soon be detected via the redshifted 21-cm line from neutral hydrogen, allowing us to indirectly study the properties of HMXBs in the unseen, first galaxies. Low-redshift observations, as well as theoretical models, imply that the integrated X-ray luminosity to star formation rate of HMXBs ($L_{\rm X}/{\rm SFR}$) should increase in metal-poor environments, typical of early galaxies. Here we study the impact of the metallicity ($Z$) dependence of $L_{\rm X}/{\rm SFR}$ during the EoH. For our fiducial models, galaxies with star formation rates of order $10^{-3} - 10^{-1}$ $M_\odot$ yr$^{-1}$ and metallicities of order $10^{-3} - 10^{-2}$ $Z_\odot$ are the dominant contributors to the X-ray background (XRB) during this period. Different $L_{\rm X}/{\rm SFR}$-$Z$ relations result in factors of $\sim$ 3 differences in these ranges, as well as in the mean IGM temperature and the large-scale 21-cm power, at a given redshift. We compute mock 21-cm observations adopting as a baseline a 1000h integration with the upcoming Square Kilometer Array (SKA), for two different $L_{\rm X}/{\rm SFR}$-$Z$ relations. We perform inference on these mock observations using the common simplification of a constant $L_{\rm X}/{\rm SFR}$, finding that constant $L_{\rm X}/{\rm SFR}$ models can recover the IGM evolution of the more complicated $L_{\rm X}/{\rm SFR}$-$Z$ simulations only during the EoH. At $z<10$, where the typical galaxies are more polluted, constant $L_{\rm X}/{\rm SFR}$ models over-predict the XRB and its relative contribution to the early stages of the reionization.