I think that has to be one of the hard things for the uconnect type devices to do. Basically cancel out the frequency from the mic that it is amplifying for the speakers. Without going high tech I don't know how you are going to get the mic too far from your mouth. If it is a decent bluetooth thing then maybe it has the brains in it for cancelling the feedback. I would try to either amplify the mic a LITTLE or possibly try other microphones. Mics for different purposes have different outputs. Possible a hign output mic that is somewhat directional???
Using an amplifier has many issues it brings to the table, like impedence matching, distortion, possibly overdriving the bluetooth input, etc.
Keep us posted.