The video would be great, if you can find a way to post it somewhere and provide a link.
When you say the rudder oscillates, does it hunt back and forth like it keeps overshooting the desired position, corrects and overshoots again, etc. with an oscillation period of around a second? Or does the heading oscillate, which is slower and a more common complaint?
Rudder oscillation can be cause by a bunch fo things. First to check is all the steering gear to be certain there is no play in any of the various joints. You will need someone to rock the steering wheel back and forth while you inspect joints. There should be no play anywhere. Even the smallest amounts can lead to rudder hunting. Also check the APs rudder feedback linkage to be sure there is zero play.
Air in the steering system can cause this too. If the pump was added/replaced as part of the installation, air in the system is a distinct possibility. Bleeding air out of a steering system can be maddening difficult for what should be a very simple task. If you have groaning sounds from the steering when the pilot is operating, it's likely air in the system.
I'm not familiar with the Garmin pilots, but the others I have dealt with have a setting for Rudder Deadband which is the smallest amount that the pump can reliably move the rudder. If the deadband is set too low, the rudder will hunt. The pilots I'm familiar with automatically set deadband as part of the commissioning process, and usually get it wrong - sometimes quite wrong - so you may need to adjust it manually. 0.75 to 1 deg is a good range. If you have to set it higher to eliminate hunting, then you probably have play or air in the system. If you can get it below 0.75, power too you. Lower is better and reflects finer rudder position control.
Another factor directly linked to deadband is the steering pump capacity, which translates into how fast it can move the rudder. Fast is good when maneuvering, but fast also increases deadband. With too fast a pump, but the time the pilot has commanded a rudder movement, the pump has pushed the rudder too far, and that leads directly to hunting. A slower pump allows fro lower deadband. You need to find an acceptable middle ground between rudder movement that is fast enough for maneuvering, and slow enough to get acceptable deadband. I have found around 12 seconds stop to stop rudder movement is at least a good starting point.