What you want to do, is to enable the "Global Hotkeys" option in Livesplit's settings. With this, whenever you press "0" (or whatever button you assigned to do the splits), it will always affect Livesplit, even if not focused.

Unrelated, you can also enable the option "Deactivate for other programs" so you won't have conflicts in the buttons. Just keep in mind that with this option, the buttons won't work at all for anything unless Livesplit is closed.