I've built cages out of AC or BC sanded with decent success.
But I found the extra money on birch or oak laminated plywood was well worth the extra $10 per sheet or so. The wood is straighter and gives you two smooth surfaces. With AC or BC the C side will have a lot knots and the sheet will not be as straight as oak or birch laminated plywood.
I've actually seen some decent CDX, but I've never been sure about the treatment and I've also heard it is heavier than other plywoods. Makes sense that it would be heavier if chemical treatments are indeed used.