I have read a couple of comments and am a little confused. If you added 1/2" tileboard, and another 1/2" for tile, the transition would be one full inch. Don't get me wrong, but the 1 1/4" sound a bit higher than it should be. If you had 1/2" carpet pad, and 1/2" carpeting, you'd be pretty close without any subflooring.
I definitely would not use shims.
I also don't think you'll get anywhere with the builder. A flooring budget is a flooring budget, and it isn't right to expect the builder to install a subfloor, unless it is called out ahead of time.
The best solution is to be very careful about the exact heights after carpet and pad are installed, and make up the difference with some inexpensive filler board, probably orient strand board (OSB). The board isn't going to serve any purpose, other than a huge shim, so I wouldn't be too worried about its quality, and cheap plywood might actually have voids versus OSB (chip board), but make sure it is installed very well, probably glued and screwed to ensure no squeaks.