Thought of a fairly major shift of approach last night around 1:30. Woke up this morning with my brain still buzzing about it. Dragged myself out of bed and sat down in front of the whiteboard for an hour or two of frantic scribbling and occasional cursing. Turns out the crazy late-night idea doesn't work at all, but something derived from it looks terribly promising. Mostly because in imitating the things I'm trying to encode judgmentally I'm no longer doing what amounts to inversion on non-invertible rules --- which I was doing before, which I think is the root of the cut-elimination-proof failure.
So I'm still without an example that really tests the key feature of the new system: I need to find a proposition A such that ~~A^o is provable (in the empty contex) but ~~A is not, where ^o is the double-negation translation (A => B)^o = ~~A => ~~B (A v B)^o = ~~A v ~~B (A ^ B)^o = ~~A ^ ~~B p^o = p T^o = T F^o = F