Note that if you have ShopSite Pro and know JavaScript (or know/hire someone who does) then you could use the Shopping Cart JavaScript Variables feature to determine the total weight of the order.
http://www.shopsite.com/help/8.3/en-US/sc/pro/cart.variables.html
The ss_weight JavaScript array created by this feature will contain the weight of each item in the cart in each element of the array, so if you loop through this array in your own custom JavaScript and add up all of the the weights then you would have the total order weight, which you could then place somewhere on the shopping cart screen using a JavaScript document.write. Using a default ShopSite shopping cart template you could place your custom JavaScript at the top or bottom of the screen using the 'Text at the top of the..' and 'Text at the bottom of the..' fields under the various submenus of Commerce Setup -> Order System, or if you are using a custom shopping cart template then you could put it almost anywhere in the shopping cart screens that you want.
-Loren