No. That's not how Bluetooth FM transmitters work.
1. Plug the FM transmitter into the cigarette lighter socket
2. Chose the FM transmitter, in your Bluetooth device list on your smartphone and establish a bluetooth connection(or Watch, if it could find the FM transmitter. It works with my iPhone or any other Smartphone)
3. Chose any free Radio channel on the FM transmitter
4. Chose this radio channel on your car radio and enjoy your music.
So, this works since many years with any smartphone. The current problem is, that the FM transmitter doesn't show up in the BT device list of the Watch. There is also a similar problem with some car dash, which won't pair with the Apple Watch. There are obviously some BT restrictions on the Apple Watch. So, i'm searching for a FM transmitter which can establish a BT connection with the Watch.
(Smartphone: sends audio data via BT --> FM Transmitter: receives audio data via BT --> FM transmitter emits FM signal --> Car radio receives FM signal)