Note that there are "hacks" ways to add Apple CarPlay to the Tesla's screen. They aren't great, but they do function. And Lyft Driver is available as a CarPlay app. But it only shows the map/navigation, no other interaction possible.
My only CarPlay vehicle is one I can't use for Lyft, so I can't test if it even pops up the "Accept Ride" notification or not.
Not sure it's worth any of the (not cheap) CarPlay hacks.
There are also "software-based screen mirroring" apps like TslaMirror, but those have no input, they just show what is on your phone's screen. You can't tap the "Accept ride" button on the Tesla's screen with it. Those also all cost money, although less than a hardware CarPlay hack.