Lets start all over.........
Kitchen sink clogging.......but 'P' trap clear and snake down 6' clear.
So what do you mean by clogging?
When you go out to your 'service center' where you hook up city water, connect sewer hose and have dump valves for holding tanks.......how many valves?
Do you have a clear adapter (after market purchase) connected before sewer hose so you can see what/how draining is going when you open dump valves?
I'm assuming you are having backup problems in kitchen sink cause grey tank is not being drained and is full......over full.
Galley..that would be a second grey holding tank that kitchen sink (galley) only drains into and will have it's own drain valve. Usually within close proximity to kitchen sink (outside)......so IF and big IF you do have a galley that valve might be in another compartment separate from service center.