I set one of my extra mouse buttons (Elecom Deft Pro) to ⇧⌃⌘U using their Elecom Mouse Utility. The "Click here to record a shortcut:" picks up the correct input, but will not work outside the record input. The command will work with a normal keyboard just not the mouse button.
My current workaround was to create the action as a Menubar Item to which I added a keyboard shortcut using Bartender 4 (I am also able to use this button (⇧⌃⌘U) with the system keyboard shortcuts, and apple shortcuts). Seeing as other apps are able to use this button as a shortcut it seems like a bug with BTT.