Afraid not. What you've got is the "new and improved" outboard bearing type BB. I don't think you can use a bottom bracket spacer with it. If it was my crankset I'd replace the 34 with a 36 because the 36 would be more useful to me and I believe I can climb what I want to climb with a 36 in the front. I can't guarantee that going to a 36 would completely solve your problem but it should give the chain a little more clearance on the big ring. I've heard of some people solving the rubbing by spacing out the big ring with thin spacers added to the chainring bolts. But of course this will also space the two rings farther apart.
I have never worked with American Classic cassettes, not sure exactly what you have but would caution that removing a cassette spacer may not allow the lockring to tighten the cogs down enough, not saying that you can't try that, but I do recommend that you work with the crankset first.