Paul,
Are you sure about the side play at the breach bolt

Does it make any noise, or can you feel a 'clik' when you move the barrel back and forth sideways

?
With only light pressure from one finger I can get the muzzle of my Whisper to move sideways 1/8 inch easily. As you may know, with the "thing" on the end of the barrel it makes a Whisper barrel rather long, giving increased leverage/easier bending. And the barrels on Whispers are just a thin steel sleeve inside plastic. That added length, plus thin tube makes it quite 'whippy' as gun barrels go.
When I cock the gun holding the barrel down, then look down the barrel, I can see quite a bit of the top plastic edge of the muzzel "thing". But when I bring the barrel back up to where it will stay open without my holding it, then look in the breach, no edges of the muzzel "thing" are visable. That tells me the barrel is bending with not much pressure, but when the bending pressure is released, it goes back to where it should, which is OK.
Early on with the gun, I thought the same as you. I measured the side clearance at the breach bolt with feeler gauges - about .0075 on each side. I then made brass shims to slip in and take up all but about .001 of the play. Made no differance in accuracy so I took them out.
Turns out the accuracy issues were that the seal was probably damaged during original assembly and was going bad. In addition to that, a bad Gamo scope, and then problems with the new scope's one-piece mount stop pin. All that's taken care of now, the barrel is still just as 'whippy', but at 15 yards the gun puts Exact Jumbos into one ragged hole.
Paul.