On my Hohner Style O copy, the bridge sits on a six-legged cobweb-like structure of fairly substantial metal, which in turn rests on the rim of the cone, and is connected to the point of the cone by a long screw type of thing. This would give you some latitude to move the position of the bridge slightly without causing any permanent and substantial structural changes. Or is yours not constructed that way?

Incidentally, I don't have any such problem with tuning, except on the first string around fret 10 or 11, where it "skips" a fret. Shame I'm such a crap guitarist!

Steve