Postal tip that few know: If an item is not heavy, you can pack it in any box, they will weigh it, and put priority mail tape on it. You will still get the insurance and tracking, and often is cheaper than the flat rate. Hope this helps. :)
But this will obviously be heavy and it will take a sizeable box, anyway; flat rate, or not. A large flat rate box would be at least 12 dollars or so and that would be at some place which might not claim to specialize in packing. Something like this has got to be packaged with TLC