I had the same problem on my S7 and I finally solved it!
I had this problem with the exchange account on my S7. Contacts were not available on the car-kit through Bluetooth and I couldn't save a new contact to my exchange account.
Seems that Samsung removed some functionality from Android, while it is still functioning underneath.
To solve it I installed the standard Google Contact APK which you can find here:
Google Contacts APKs - APKMirror
(This isn't officially compatible, but I didn't experience any problems so far)
Now it is possible to save contacts to the Exchange account and set the default account.
In the settings you can found an option: "Visible Contacts." Choose change and you will be able to select the contacts in your Exchange account contacts by label / all other contacts. At mine there is an option: "Share profile" at the phone number which I checked as well for all contacts.
After I changed this settings the car (in my case BMW 5 / 528i / F10) recognized the contacts again.
It took me a while to figure this out. Tried several different firmwares, but this seems to solve it.