The upcoming radio interferometer Square Kilometre Array (SKA) is expected to directly detect the redshifted 21-cm signal from the neutral hydrogen present during the Cosmic Dawn. Temperature fluctuations from X-ray heating of the neutral intergalactic medium can dominate the fluctuations in the 21-cm signal from this time. This heating depends on the abundance, clustering, and properties of the X-ray sources present, which remain highly uncertain. We present a suite of three new large-volume, 349 Mpc a side, fully numerical radiative transfer simulations including QSO-like sources, extending the work previously presented in Ross et al. (2017). The results show that our QSOs have a modest contribution to the heating budget, yet significantly impact the 21-cm signal. Initially, the power spectrum is boosted on large scales by heating from the biased QSO-like sources, before decreasing on all scales. Fluctuations from images of the 21-cm signal with resolutions corresponding to SKA1-Low at the appropriate redshifts are well above the expected noise for deep integrations, indicating that imaging could be feasible for all the X-ray source models considered. The most notable contribution of the QSOs is a dramatic increase in non-Gaussianity of the signal, as measured by the skewness and kurtosis of the 21-cm probability distribution functions. However, in the case of late Lyman-alpha saturation, this non-Gaussianity could be dramatically decreased particularly when heating occurs earlier. We conclude that increased non-Gaussianity is a promising signature of rare X-ray sources at this time, provided that Lyman-a saturation occurs before heating dominates the 21-cm signal.