I notice that devices that work are only speakers, which use the BT A2DP profile (audio stream), whereas with the car player, your device presumably opens a BT AVRCP profile (control from the car's controls) as well.

That matches my experience: when playback is controlled from MMA, the BT connection usually functions just fine; but connecting to my car player with AVRCP, and I get several varieties of "won't play reliably or at all."

This may be related to the version of Bluetooth on your device, and the implementation of the car's BT, as well. My device is stuck at Android 4.2.2 and so missed out on whatever BT improvements there have been in Android since then. BT worked a lot better for me with Android 4.0 and 4.1.

Environment:
HTC One M8
Android 6.0
MM Gold 1.1.3.0554

Issue:
MM will not save the song position when disconnected from certain BT media players. This feature worked with a previous version of MM on all players (sorry, dont know exact build number, but late 2015 build). The audiobook resume works on all players, but the music resume does not work with some players.

Tested working players:
Mini-Jambox
Large Jambox
Logitech bluetooth adapter

Tested not working players:
Pioneer MVH-x380BT

Reproduce:
Car is turned on, media monkey connects and starts playing music. When power to the car stereo is turned off, a message on the phone will say 'bluetooth device disconnected"' and the track progress bar will return to 0%. On reconnect, MM will start playing the song from the beginning. ON other players on disconnect, MM will save the position of the song.

Attempted Fixes:
Upgraded Android to 6.0
Upgraded MM to latest version
Various options is MM.

Again, this seemed to be working in previous versions but the latest builds seem to have broken it. Not really sure if it is expected behavior or not.