One way to achieve this is to create premade classes with background colors, e.g., red-bg, blue-bg, black-bg, then assign the background colors via CSS. You can then add the respective class into the Heading Module and get the background color added. Your client should be able to control the color of the heading page by page this way.
It does make sense, but in this case, we aren’t going to know what the page colors will be until they create the page. So I’d end up making custom classes every time they wanted a new page.
I can setup a different style as a workaround for now. It would be nice if we could have options on the heading module to add a background and then set it to full width (like the buttons). Consider this my feature request.