I would like to suggest 5 new settings in the admin panel at the appropriate places.
ShowCartItemWeight, bool, to show the item line weight in the cart (product+variant weights etc x qty).
ShowCartItemWeightAdmin as above functionality for any carts viewed in the admin site, ie sales->current carts etc.
ShowCartTotalWeight, bool, to show the total cart weight in the cart summary block.
ShowCartTotalWeightAdmin as above functionality for any carts viewed in the admin site, ie sales->current carts etc.
ShowProductWeight, bool, to show the product item weight, dynamically updated when attributes are chosen/selected.
Most other ecomm platform s support this for a better customer shopping experience, and also then allow s customer to contact store owner for a custom special shipping manual quote price based on weight of cart etc. Opens up other feature opportunities also.