The Steam Deck compatibility notes for the game specifically say that it isn’t good at handling multiple controllers:

This game does not default to external Bluetooth/USB controllers on Deck, and may require manually switching the active controller via the Quick Access Menu

I think it is likely that this problem comes from the game, not a problem with streaming sending the inputs. You can try testing this in another game with more flexible controller support to narrow it down.

In case it is helpful: Big Picture Mode does show non-Steam games. Go to the library and there is a tab for them. They’ll also show in the recent games list on the home page.