Is there a setting in the car radio to stop automatic connection? What I do in this situation is switch the car radio off after I start the engine before it has a chance to grab the Bluetooth connection as that is easier to do than burying into the menu settings to turn automatic connection off then back on again afterwards. But mine is an old radio in an even older car and I know modern cars with modern integrated ICE systems take that level of user control away in my experience.
The other option is to unpair the car radio from your phone but that can also be a faff if you are normally wanting to use it. I guess, like me, it is for those short journeys when you want a continuous listening experience that you want to stick with the ear buds.