@John Pym You can workaround this limitation with a custom keyboard shortcut. For example in gnome. Going to settings-keyboard shortcuts. Scroll down and press +. Add a name "foobar2000 play/pause" in command add
env WINEPREFIX="path to your wine prefix" wine path to foobar2000.exe /playpause in shortcut choose a key combination (ctrl+alt+p) for example.
You can also use the original media keys but you must disable default shortcuts and you will lose the gnome integration (mpris, etc)
With foobar2000 running this shortcuts works even if it is not selected. Problem is that if is not running the shortcut start the player instead of fails.
Same for the rest of foobar commands (/stop /exit etc)
Is not perfect but usable