Hi David,
The hard part about that is that at the time the credit card is initially charged, you're going to get an authorization for the initial amount. From my experience, you can't capture a payment that's larger than the original amount.
Depending on your credit card payment gateway, you may be able to have your shoppers safely and securely store their credit card info. This info does not get stored in Magento, but rather, is stored in a vaulting system with your gateway, and is tokenized. Solutions like Authorize.net CIM are popular for this, and there are Magento Extensions that will allow you to leverage them, such as from ParadoxLabs.
The benefit here is that with the credit card on file, you can potentially either cancel the original order and authorization and create a new one with the surcharge included or just add a new authorization for the additional amount. Either way, you'd be creating a bit of a unique flow.
If you're looking for an off-the-shelf solution to test with There are Magento extensions that make it easier to edit an order, such as:
https://marketplace.magento.com/cminds-cminds-orderedit.html
https://marketplace.magento.com/aheadgroups-aheadgroups-ordereditor.html
Feel free to review to see if one can meet your requirements.
Additionally, you might also consider a workflow where a user requests a quote, and you submit back an estimate for the shopper to approve and pay. This way, there's only one authorization. There are extensions for this like https://marketplace.magento.com/cart2quote-cart2quote-customer-quotation.html.
Best of luck!
