I had the same problem that many have reported concerning getting my Droid Bionic to transfer the phone's Contact List to my in-dash JVC Navigation Unit. I solved it by reading the JVC Manual -- which gave one hint of what the problem was (not verified). And reading as many Posts on the Web as I could find -- most of them negative about the Droid, the Bionic and Motorola.

Here's how I did it. Hopefully, it will work for you:
1) Go through the steps of Pairing your Droid Bionic to the in-dash JVC unit. Oddly enough, this can, apparently, be done with the vehicle running or without -- so long as there is power to the JVC.

2) Now, you'll discover that the Droid Bionic has no "Send" option for transferring the Phonebook.

3) The trick seems to be -- for the JVC (and maybe other brands) -- is to do the following with the engine turned OFF. Why? Because somewhere I came across a statement that transferring of contacts while the car is running is not allowed "for safety reasons". Aha! methinks.

4) Turn the vehicle OFF. Then turn on your Phone if it's not already on. Be sure the Bluetooth indicator appears. Turn the ignition key to the first position which allows your JVC in-dash unit to come on without starting the engine.

5) Watch your phone at the top of the screen. As the JVC Bluetooth connection is established a message will appear reading "JVCNavi is requesting phonebook information" -- tap on that message. A window should pop up on the phone asking if you wish to "Allow" or "Always allow" this. Choose either (I forget the exact wording).

6) Voila! The JVC unit should show the connection to the device and the phonebook transfer status. Mine took all of 5 seconds. After that, you can proceed with editing your Contact List on the JVC (or whichever in-dash radio / stereo you might have).

For the record, the latest JVC units are advertised as "Requires PBAP-equipped phone." That had me worried because Bluetooth File Transfer (an App suggested in this Forum) reported that my phone was "Not PBAP-equipped". However, the JVC Owner's Manual for my model clearly states it supports the following Bluetooth devices. Those equipped with:
HFP 1.5
OPP 1.1
PBAP 1.0
A2DP 1.2
AVRCP 1.3

Maybe my tips posted here will help. I cannot swear that my steps, above, are mandatory. All I know is when I tried to do this with the car running, it did not seem to work. You might try simply looking for the message at the top of the phone's screen when you first start the car. Navigate your in-dash unit to the phone or Bluetooth tab. Then tap the message on the phone and allow the response. That message disappears fairly quickly, however.

Oddly enough, with all the complaining I was reading online about how difficult this was, it ended up being incredibly simple. Just not well-explained in any literature -- not the Bionic's Manual, nor the JVC Manual.

Good Luck!