Flume does not offer a custom full-screen experience for various reasons. You may however use the macOS Split-View feature (macOS 10.11 or later) to use Flume and another application side-by-side without compromising the user interface.
To activate Split View mode:
Hold the ⌥ OPTION key while long-pressing/clicking on the main window's zoom button (the green button in the title bar area)
You will be able to choose on which side of your display you would like to keep Flume.
Select another application that will be displayed beside Flume in Split-View full-screen mode.