High Dynamic Range (HDR) videos have enjoyed a surge in popularity in recent years due to their ability to represent a wider range of contrast and color than Standard Dynamic Range (SDR) videos. Although HDR video capture has seen increasing popularity because of recent flagship mobile phones such as Apple iPhones, Google Pixels, and Samsung Galaxy phones, a broad swath of consumers still utilize legacy SDR displays that are unable to display HDR videos. As result, HDR videos must be processed, i.e., tone-mapped, before streaming to a large section of SDR-capable video consumers. However, server-side tone-mapping involves automating decisions regarding the choices of tone-mapping operators (TMOs) and their parameters to yield high-fidelity outputs. Moreover, these choices must be balanced against the effects of lossy compression, which is ubiquitous in streaming scenarios. In this work, we develop a novel, efficient model of objective video quality named Cut-FUNQUE that is able to accurately predict the visual quality of tone-mapped and compressed HDR videos. Finally, we evaluate Cut-FUNQUE on a large-scale crowdsourced database of such videos and show that it achieves state-of-the-art accuracy.