Is the problem you are having just the "display" of the Weight in the Cart?
If so, then you will need to customize the:
and change how the weight is calculated:
NOTE: many of your Products have Attributes and if they are needed to buy the Product, you should add an Option Value such as:
// build shipping with Tare included
// $shipping_weight = $_SESSION['cart']->show_weight();
$shipping_weight = $_SESSION['cart']->show_weight() + $_SESSION['cart']->free_shipping_weight;
Display Only: YES
Sort Order: 0
so that the customer has to make a selection ...