The problem is that once the purchase is done, you can't modify it. You would've have to either refund the some money or charge some more. Or you can use apps I mentioned above, where you can (supposedly) edit the order and modify it's price quantity...

I did some research and found 2 apps which might solve your problems.Edit Order by CleverificEdit Orders - EditifyIf these apps doesn't solve your problems, let me know, so I can continue working on a customized solution for you.

Currently working on a solution. Should be live in a month or so. I'll keep you updated.

But you would still have to charge 16$ initially no?  And then based on the actual weight you would either refund them a few cents or charge some more? If that's not the case, you would have to delay the payment until you know the actual weight.