Randoid, BT streaming like that is something that I have never tried, so I probably won't be much help. That said, I will say that I sincerely doubt that rooting will solve the problem. As far as I know, rooted phones and custom ROMs for the D3 that work with Bluetooth are just using the same BT stack as the stock OS.
In scanning through the D3 and Bionic threads (the Bionic had a very similar build as the D3 when it had Gingerbread), I've seen the same issue reported without resolution. (See this, for example: http://androidforums.com/motorola-droid-bionic/414463-bluetooth-issue-jvc-kd-r820bt.html
This may sound weird, but is it possible that there is a software/firmware update for the JVC deck? I'm not sure if your model has an update, but this may be the place to look: Software Updates
If you look at that page, for example, there is this one:
KW-ADV64BTJ, KW-AV60BTJ, KW-AV60J, KW-AV68BTJW, KW-AV70BTJ, KW-AV78BTJW firmware update
This update improves the Bluetooth compatibility with CDMA phones.
The Droid 3 is definitely a CDMA phone... obviously, that may not be your deck, but perhaps there is an update?