CDPR is eager to move on from the Red Engine, explaining why Phantom Liberty is the only Cyberpunk 2077 DLC.
Not sure where you’ve got it bought, but I pretty much installed it using Heroic (I have it on GOG), then added it to Steam as a non-Steam game (with Heroic it’s a one click operation) and then simply found a Steam Input config that works well. I can check the config name if you want.