Okay, so I found that there is a "Set variation shipping" link under each variation when you set up a Product. Clicking this link allows you to set up shipping costs for each variation.
One thing to note though is that this value is added to whatever you entered as the Base Shipping in the Store Settings for Shipping. So to get an accurate shipping cost in my example above, I made the Base Shipping for everything be the minimal cost of 6 and 10 (the unstretched cost). Then for each Stretched Variation I set the shipping to 6 for local and 10 for international. So this totals up on the checkout page to 12 and 20, just as required. Sweet!
Another point to note is when you purchase 2 or more items, the shipping cost of the specific variation is what gets added on. So, for e.g.
1 stretched item = 12 shipping cost
2 stretched items = 18 shipping cost
4 stretched items = 24 shipping cost and so on.
Hope this is helpful to someone.