You have to pack it really tight with bubblewrap. There has to be enough to support the cooler, otherwise I don't see how it's going to help at all. Personally, I would probably fashion something out of cardboard. There's less give in that than bubblewrap.
Anyway, if it's with a forwarding company that likely has no clue / little interest in what you're asking them to do, I would request it to be shipped in a large enough box filled with packing peanuts, and then pray to the shipping gods. Packing peanuts will dampen any hard drops and hopefully keep the cooler from being knocked out.