sure you can show/hide it via any trigger in BTT.
Basically change it from "Visible on Launch" to "Visibility Toggled Via Action". Then assign the predefined action "Toggle Floating Menu Hidden/Shown" to a keyboard shortcut, gesture or anything else.
For more advanced use cases you can also use the separate "Hide Floating Menu" and "Show Floating Menu" actions (e.g. if you would like to show it when doing a three finger click on the trackpad and hide it when removing the last finger from the trackpad).
I'm currently working on the documentation and will add a page about how to trigger floating menus.