Big Apples never sit entirely straight on the rim, but I don't think that's what the OP is talking about. I'm guessing there is a cut, tear, or some other dislocation inside the tire casing. It's probably more identifiable inside the tire than on the outside, but it may not necessarily be a clearly visible problem on either side.
Whatever it is, if it's not related to the depth the bead is seated on the rim, it's probably not fixable.
I wouldn't use the Big Apple in a size and width that you can get regular freestyle tires in. It's more expensive, and not any faster or tougher than a lot of them. I'd use something cheap, fast, and reasonably tough like the CST Operative or Animal GLH, or the Primo Comet if I wanted something a little faster yet. There are now a lot of very fast, lightweight yet streetworthy folding tires for weight weenie freestyle bikes, and most of them are in the same general price range as Big Apples.
As a tangentially related note, the most heavily armored tire setup I have ever furnished to a customer was a pair of 20x1.75" CST Operatives with the bead wires cut off, tucked in as liners to a pair of 20x2.25" CST Operatives. (This is what New Mexicans call "the system"-- a second tire used as a tire liner.) The whole setup was less expensive than a pair of Big Apples, and it didn't ride nearly as badly as you'd think.
This is to express my gratitude to Justin of Grin Technologies for his extraordinary measures to save this forum for the benefit of all.