First question, what do you mean by "burp" the grey tank?
Here is a generic drain plumbing diagram that may help you visualize where the problem may be.
The grey tank is not only vented through the roof, but each drain line from a sink has an air admittance valve (AAV) that allows air into the drain line when negative pressure from running water exists. These valves are on top the the grey water vent by the sink in the diagram.
Again, your coach is probably not plumbed exactly like the diagram but you can see the shower is vented through the main vent through the roof. You may also not have a P-trap but a Hepvo waterless waste valve instead. If only the shower is draining slowly, or not at all, and the main vent is located between
the shower and the grey tank, it is possible there is a blockage to the main vent which is not affecting the other drains due to the presence of AAVs in their drain lines.
Your coach may have an AAV in the shower line as well, but it would/should be is some accessible space between the shower drain and the grey tank. Most coach builders don't provide the access nor put AAVs in the shower line.
Again, sorry I don't know what "burp" the grey tank means but whatever you do seems to open the shower line for drainage. It may be possible the main vent pipe was installed too far into the grey tank, or has slipped down into the tank. However, since the draining issue happens even when the tank is empty, that possibility seems remote.
Whatever you do, don't push anything like a snake down the shower line unless you know for certain you do not
have a Hepvo waste valve. Water from a hose, flowing down the main vent pipe from the roof, should go straight into the grey tank with no problem.