Unfortunately, DOMRenderer doesn't have the capabilities to create a full 3D rendering, as it's limited to applying z-indexes and scale. This means that creating a rotating cube using DOMRenderer is not possible.
One alternative solution could be using html2canvas to generate a bitmap texture that can be applied to the cube. While I personally haven't explored html2canvas extensively, the demos showcase some impressive results. It's worth experimenting with to see if it suits your needs.
If you're looking for a more suitable option for your project, CSS3 3D transforms might be a better fit. There's a library called traqball that can assist with this type of task, which you can find at https://github.com/dirkweber/traqball.js.
For a demonstration of CSS3 3D transforms in action, you can check out this demo: