Dennis,
How much movement, and are you sure that's what's causing the accuracy problem? Could it be that the scope or mount has moved ? Or the seal is going

?
My Whisper has had about .007 to .008 inch clearance on either side of the barrel pivot point since day one. If I "slightly" push/pull sideways on the barrel it will also move. Some of the movement is actually the barrel bending. As you can see on yours, the barrel is just a thin steel tube inside a plastic covering.
When I first got the gun, I thought that much side clearance would be a problem so I made brass shims and slipped them in to bring the side clearance down to about .001 each side. Didn't make a bit of differance in accuracy.
When I started checking the gun, I found that the Leapers one-piece Acushot scope mount had sheared the 1/8 inch diameter stop pin and oval-ed the stop pin hole in the reciever. I made a 1/4 inch diameter stop pin so that it would be stronger, and have more surface area to bear against. Then I drilled out the reciever to very closly match the size of the stop pin diameter. The scope hasn't moved since.
Shortly after that, the seal gave out and the accuracy was inconsistant again until I replaced it. Now, hundreds of pellets later, with the over-sized stop pin and a new Maccari seal, it's been very consistant !!!!!
Paul.