The "ships it fits" boxes have a weight limit of up to 70 lbs I believe. If it fits in the box, it's the price of the size box you have up to the weight limit. Hence why they're called flat rate. You could put your 6"x6" box inside the medium box with packing material, and it's still $18. I double box things all of the time for extra protection.
I ship things multiple times a week, and rarely get $25 shipping charges in the lower 48. I usually only exceed $25 because of the oversize box fees. It's your part, your process. Good luck.