I moved this thread to its appropriate forum section, this topic is not Launchbox issue.
Launchbox has no settings or controls over your controls in emulator, controls are all set up in your emulator, controllers in LB only control LB itself.
You have several layers of potential issues going on here with Steam BigPicture and InputMapper it is hard to know exactly where your issue could be.
I have 2 controllers hooked up to my PC and I have to tell RA what controller I want used with what core and save core overrides. One controller is an Xbox One controller and the other is a Hori FC, both are wired and Xinput.
If your controllers are wireless or you are unplugging controllers and plugging in others there will be issues, this is a known Windows issue and how it assigns IDs to the controllers.