Android Transmit system audio over MTP
No, MTP doesn't work like that. MTP is a client-server file transfer protocol, a bit like FTP. Your best bet is to either find another protocol the car supports (there are many network audio protocols, so it's likely it'll support one), or failing that, fix whatever is making the aux input sound bad (most likely a bad cable or connector, or the car is expecting line-level instead of headphone-level audio).