A seven pound 416! That would give me a headache quickly.
My .375 African usually sits in a HS Precision stock that I ordered from Stocky's. It did not reduce the weight of the rifle but is a good stiff synthetic that is weatherproof and has a good recoil pad. It is the 23.5" barrel model and it weighs around 9.5 pounds with scope and sling.
I like the idea of a short barrel on a .416, but for my purposes it needs a walnut or laminated classic style stock, a good recoil pad, and a weight somewhere between 9 and 10 pounds.