I personally would like to see you keep the batteries as a 'damper' so to speak and also if the charging system fails, you can still use the radio for a while.
I would have the radio on charge from the gyro constantly so the batteries aren't responsible for all power to the radio
I just think that it would be easier on the radio if the battery can absorb any spikes in voltage, or make up for any negative spikes....
I have one of those voltage reducers.... mine has a switch so you can set it on 7.5v, 9v or 12v