Wore out three sets of tires already!
I've had the same thing happen. One thing I've used is a more "toothy" washer. A problem that can happen with the 3-speeds is that a small shift in the wheel position, even before it rubs, can cause the shifter indicator chain to get out of adjustment and you can get in between gears and freewheel or grind. Remember the cable is slack in third gear and if you put the shifter into third and have a slack cable you can still ride home if you don't have the tools to move the wheel. A bit of a tougher ride but better than grinding. The Sheldon Brown site is loaded with good info on the Sturmey Archer hubs. Here's a link to the indicator chain adjustment,