Sounds like a case of round-hole square-pegging. How nice is your frame? Maybe you can have the dropouts cold set (fancy word for bent) so that the original axle will work?
The reason shims won't work is because the axle is steel. Thin sheets of AL are way too soft. If you can't bend the stays, get a reproduction of the original axle made by a machine shop. It won't be cheap, but that's the only other way I can think of.
Unless you can find a 135mm hub that uses the same size of bearings as the BMX hub, with the same spaceing as well, I don't see any other ways out of this.