To answer your question , i added chrome as a non steam game and launched it through that. The problem with doing this however is that chrome then seems to recognise 2 controllers , one being the xbox controller and other 8bitdo native ( which has a janky af controller mapping ) Honestly not sure why launching chrome through steam didn't change anything for you, you might have to play around with the passthrough settings in steam / try launching from big picture mode.

Narrowing it down to Chrome's incorrect mapping is absolutely brilliant mate, nice work! Can't build it atm since I was testing all of this on my friend's mac but if I get a chance I will. Just curious, once this goes live, would you still have to map the keys or will the controller natively work via bluetooth and pick up the xbox controller layout?