So, after lots of testing today and help from people killing those who were being jerks to me and someone even showing me other spawn spots, we now have conclusions! 😛
Anywhere that the Ancient Gree Destroyer bot can spawn is where you want to be. Besides the giant dome in the center of the PVP area, it can also spawn in the Southern area in various places.
The robot spawns every 7 minutes.
On rare occasions, instead of an Ancient Gree Destroyer spawning, an Ancient Gree Defender will spawn instead. This robot is a lot smaller than the giant Destroyer Droid–similar to the war bots patrolling around and is about the size of a character.
While Ancient Gree Destroyer will always drop the small reputation, Ancient Gree Defender ALWAYS drops L1-L Defender. Please keep in mind that the bonus quest does ask you to defeat an Ancient Green Destroyer–not a Defender so defeating an Ancient Gree Defender, despite it spawning in the same place (and thus, if one is up, it needs to be defeated before a Destroyer will spawn), will not complete the bonus quest.