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.

Base game, no. The Adrenaline Pack Pt 1 DLC has rallycross tracks, a few dirt circuits, and a few classes of cars for these.

That is a massive nerf, but I don’t think it’ll stop me from picking it.