Bottom line on using a separate ground rod near an antennae that is not bonded to the main house ground is, be prepared for much more damage and ruined electronics in the house if you get a lightning strike. A difference in ground potential between different parts of the house is probably more responsible for damage than not.
Right now if the house gets hit the AC mains and associated ground will be at a close potential to each other and will raise above the surrounding ground by potentially thousands of volts during the strike, but it may not be that bad between ground and the AC mains. With a separate ground rod at the antenna near the radio you can easily have thousands of volts between the AC mains fed from the house and the distant ground rod, so say goodbye to anything connected to that.
Right now if the house gets hit the AC mains and associated ground will be at a close potential to each other and will raise above the surrounding ground by potentially thousands of volts during the strike, but it may not be that bad between ground and the AC mains. With a separate ground rod at the antenna near the radio you can easily have thousands of volts between the AC mains fed from the house and the distant ground rod, so say goodbye to anything connected to that.