Here's how to get weight & dimensions for stockable variants passed into the cart! 
Open plugins/vmcustom/stockable/stockable.php
REPLACE THIS CODE:
public function plgVmOnAddToCart(&$product){
$customPlugin = JRequest::getVar('customPlugin',0);
if ($customPlugin) {
$db = JFactory::getDBO();
$query = 'SELECT C.* , field.*
FROM `#__virtuemart_customs` AS C
LEFT JOIN `#__virtuemart_product_customfields` AS field ON C.`virtuemart_custom_id` = field.`virtuemart_custom_id`
WHERE `virtuemart_product_id` =' . $product->virtuemart_product_id.' and `custom_element`="'.$this->_name.'"';
$query .=' and is_cart_attribute = 1';
$db->setQuery($query);
$productCustomsPrice = $db->loadObject();
if (!$productCustomsPrice) return null;
// if ( !in_array($this->_name,$customPlugin[$productCustomsPrice->virtuemart_custom_id]) ) return false;
$selected = $customPlugin[$productCustomsPrice->virtuemart_customfield_id]['stockable']['child_id'];
if (!$child = $this->plgVmCalculateCustomVariant($product, $productCustomsPrice,$selected) ) return false;
if ($child->product_sku)
$product->product_sku = $child->product_sku;
if ($child->product_name)
$product->product_name = $child->product_name;
$product->product_in_stock = $child->product_in_stock;
}
}
WITH THIS CODE:
public function plgVmOnAddToCart(&$product){
$customPlugin = JRequest::getVar('customPlugin',0);
if ($customPlugin) {
$db = JFactory::getDBO();
$query = 'SELECT C.* , field.*
FROM `#__virtuemart_customs` AS C
LEFT JOIN `#__virtuemart_product_customfields` AS field ON C.`virtuemart_custom_id` = field.`virtuemart_custom_id`
WHERE `virtuemart_product_id` =' . $product->virtuemart_product_id.' and `custom_element`="'.$this->_name.'"';
$query .=' and is_cart_attribute = 1';
$db->setQuery($query);
$productCustomsPrice = $db->loadObject();
if (!$productCustomsPrice) return null;
// if ( !in_array($this->_name,$customPlugin[$productCustomsPrice->virtuemart_custom_id]) ) return false;
$selected = $customPlugin[$productCustomsPrice->virtuemart_customfield_id]['stockable']['child_id'];
if (!$child = $this->plgVmCalculateCustomVariant($product, $productCustomsPrice,$selected) ) return false;
if ($child->product_sku) $product->product_sku = $child->product_sku;
if ($child->product_name) $product->product_name = $child->product_name;
$product->product_in_stock = $child->product_in_stock; // Override this even if null.
if ($child->product_weight) $product->product_weight = $child->product_weight;
if ($child ->product_weight_uom) $product->product_weight_uom = $child->product_weight_uom;
if ($child->product_length) $product->product_length = $child->product_length;
if ($child->product_width) $product->product_width = $child->product_width;
if ($child->product_height) $product->product_height = $child->product_height;
if ($child->product_lwh_uom) $product->product_lwh_uom = $child->product_lwh_uom;
// if ($child->min_order_level) $product->min_order_level = $child->min_order_level;
// if ($child->max_order_level) $product->max_order_level = $child->max_order_level;
}
}
THEN REPLACE THIS CODE:
function getValideChild($child_id ) {
$db = JFactory::getDBO();
$q = 'SELECT `product_sku`,`product_name`,`product_in_stock`,`product_ordered`,`product_availability` FROM `#__virtuemart_products` JOIN `#__virtuemart_products_'.VMLANG.'` as l using (`virtuemart_product_id`) WHERE `published`=1 and `virtuemart_product_id` ='.(int)$child_id ;
$db->setQuery($q);
$child = $db->loadObject();
if ($child) {
if ('disableit_children' === $this->stockhandle) {
$stock = $child->product_in_stock - $child->product_ordered ;
if ($stock>0)return $child ;
else return false ;
}
else return $child ;
}
return false ;
}
WITH THIS CODE:
function getValideChild($child_id ) {
$db = JFactory::getDBO();
$q = 'SELECT `product_sku`,`product_name`,`product_in_stock`,`product_ordered`,`product_availability`,`product_weight`,`product_weight_uom`,`product_length`,`product_width`,`product_height`,`product_lwh_uom` FROM `#__virtuemart_prfoducts` JOIN `#__virtuemart_products_'.VMLANG.'` as l using (`virtuemart_product_id`) WHERE `published`=1 and `virtuemart_product_id` ='.(int)$child_id ;
$db->setQuery($q);
$child = $db->loadObject();
if ($child) {
if ('disableit_children' === $this->stockhandle) {
$stock = $child->product_in_stock - $child->product_ordered ;
if ($stock>0)return $child ;
else return false ;
}
else return $child ;
}
return false ;
}
Notice I commented out a few lines in the first edited function, but you could uncomment them and add those columns (and any others) to the db query in the second function to pull more values from the child to the product added to the cart.
I suggest this be added to the core. However, it might be best if there were additional parameter options for the custom stockable variants plugin on the backend that lets you choose which values you would like the child to override for the parent.
After you make those changes your weight / dimension based shipping rules should work correctly for any new products added to the cart.
The modified default stockable.php is attached. Hope this helps someone! Took me a little while to figure it out. 

Have been through all 18 pages of this very useful thread (highest regards to the OP for taking so much time on this) and was thanked for my efforts at page 14 where the shipping weights problem I am having popped up. Unfortunately this fix is not working for the version of VM (2.0.20b) that I am using.
I am selling products with Weight variants, 1kg, 5kg, 15kg and each weight costs a different amount to ship. Of course without change, the cart only calculates the shipping weight of the parent item which is useless.
When I applied the fix about I lost my dropdown box all together so I'm guessing it need re-approaching for the newer versions of VM.
Quite surprised this has not been implemented yet. Any ideas anyone?