Seems like you have a good grasp of Polycom products, thanks for that.
Yes you can use the SoundStructure to send commands to the codec for camera positioning, to presets, via RS-232 interfaces. There is an application note posted at <here>. See section 7.7, and it applies to RealPresence Group as well as the HDX.
The EagleEye Director can be used where people are up to 22’ from the camera. See the installation guide for specifics at <here>
The issue will probably be the table is rectangular and therefore best covered with multiple cameras, which the EagleEye Director does not support. You should consider how to position cameras to get the best view of all the participants in the room. The RealPresence Group Series 700 has video input for up to 4 sources, 2 of which can be the EagleEye cameras from Polycom so you would have pan, tilt and zoom control of them.
For using a large number of Polycom cameras, up to 6, you can use Crestron or AMX control modules to control it.
I would recommend the use of a highly experienced integrator of Polycom systems to make sure the installation meets all the requirements of the Board.