Bluetooth sync with Local contacts - cyanogenmod
By chance, I found out that the local contact show up in the car when at the time before I start the car (or activate phone's bluetooth goint to the car?) I go to the Contacts App, select the options (3 dots), select (sorry german version) "Kontakte zum Anzeigen" ("show which contacts") and then, instead of "all", select the "phone only, not synchronized". 

Drive away and have the local contacts visible in the car (though not the google ones then, but most phone numbers are local here)

Additionally I found out that I can "send" the contacts from the phone to the car, but then there are too many in the phone so that the car can't hold all of them, and I'd have to send new contacts by hand...