Your first instinct of shimming behind the mount to level things out seems the most practical. You'll have the same problem mounting a 2x4 across the stone, unless you want to spend some quality time shaping the back of the board to the contours of your stone. Get a bag of fender washers so you have some surface area behind your fasteners to distribute the compression and make sure to get longer mounting screws to overcome the extra depth.
If this is a real stone fireplace, you may be well served to rent a commercial hammer drill with some muscle behind it to drill the holes for your anchors. A nice clean hole will do a much better job of holding your anchors. Speaking of which, I'd suggest something more robust than anything of plastic or tapcons, such as a wedge type
or expansion type
If your stone is some sort of cast veneer, then your plan of drilling through it to the studs is sound. Start with a masonry bit large enough to let the lag screw pass without binding, then switch to a wood bit of appropriate pilot size once you hit wood.
Once again, the lag screws that came with your mount, if any, are likely too short to overcome this extra depth. Buy some new good quality 5/16" lag screws (or larger if the mount allows it) that are long enough to sink more than 2" into wood.Edited by weaselfest - 10/7/13 at 4:03pm