Ciara here from Shopify. Thanks for posting. :)
So with the transactions on the site, you will have clearance from your customer's bank to process the amount that they agreed to at checkout.
If there's a shortfall in the amount owed due to extra shipping charges, then you can always process a draft order with the extra amount. You could perhaps create 'products' for the value of the shipping costs that are hidden on the online store, then create a draft order for that shipping cost and send the customer the invoice, with a message explaining the additional fee?
Of course the easiest way is to just ensure that shipping is fully and correctly set up for every region in your shipping settings here, it really is the best way to reduce your customers' effort, and your own bureaucratic load :)
Nice work on the site! It looks great :) Let us know if there's anything else that you need, or if there's more context here as to why an additional fee method is important for you to set up :)
I hope this helps!
Ciara G | Shopify