Looking at the info on the Amazon page (and using some common sense) it seems obvious that this package should be using the actual weight rather than volumetric weight for shipping. Volumetric weight only really kicks in for large, light objects, right?
Using the dimensions listed (45.0cm x 35.3cm x 15.5cm) gives a volumetric weight of just under 5kg which is far less than the 12.7kg shipping weight. Shipping 13kg should've cost $12.50 + 26 x $4.75 = $136, more or less like you've said.
We don't know for sure that the original box they got was packed the same, almost certainly it can't have been. However, for it to cost $241.75 means a volumetric weight of 24kg (120,000cm^3)! That sounds like a pretty big box, but it's "only" an extra 10cm on each side. I'd be really interested to know that the dimensions of the box they received was.
Edit 2: I agree that the costs are lottery but that's hardly their fault, is it? Why should they be held responsible for the varying shipping practices of Amazon et al?
Edit 3: (oops!) The problem with volumetric pricing is that just a 25% increase in each dimension doubles the weight/cost.