Have you looked into https://forums.dolphin-emu.org/Thread-unofficial-howto-using-gameini-settings-per-game? Chances are it's out of date, but it's pretty much what you're after if you're just looking to have custom settings for different games. Surely enough it's not as simple as pressing a single button though, because that wouldn't really work well.
EDIT: Uh, on the other hand if all that you're looking for is a simple way to save/restore all of your settings - go to your user directory and backup/restore the Config folder.
EDIT: Uh, on the other hand if all that you're looking for is a simple way to save/restore all of your settings - go to your user directory and backup/restore the Config folder.