Many Android phones are capable to log the full Bluetooth communication. Before Android 4.4 you needed root and an app like my Bluetooth HCI Log. Starting with Android 4.4 the log option is available on all phones in the Android developer settings.
Enable Developer Settings
Usually the developer settings are invisible. You can simply enable them by tapping 7 times on the build number in Android settings:
Open the developer menu in Android settings. You see a checkbox labelled "Enable Bluetooth HCI Snoop Log":
Start the log before you power on the car and stop the log before you send the file.
The log file is called btsnoop_hci.log and is usually stored in the root of the USB/SD storage. On some phones it is stored in the Android/data subfolder.
If you want to peek into the file you can use WireShark. Simply rename the file to *.cap and open it.