Are you logged in when that happens ? If so, did you try while logged out ?
Because I don't have an account on your website so I'm trying while logged out and it works just fine.
You mug weights 350g. When I set my address (my country is set to france), I get the shipping method "Light Package 8,00 €" which is exactly what should happen based on your shipping methods. What is the problem ?
Also, could you check that the hikashop user synchronization plugin is published because apparently the cart is lost when clicking the activation link ?