Happy to help here today but may require some more information. With the bill you have created with QuickBooks have you marked the bill 'Out of Scope'? Or is the amount you received in your banking For Review tab inclusive of GST and you want to allocate this GST somewhere else?
For the second question, if you are looking to resolve the difference and match excluding the GST you can do so with the below steps:
You can use the 'Resolve' tool when you are matching your payments as this marks the invoice as paid while accounting for the fee:
Go to the 'Banking' tab, select the relevant transaction and 'Find Match'.
Locate and select the invoice in the list. In the 'Payment' field to the right, enter the full amount of the invoice.
Scroll down to the bottom. You will see a difference which should total your fee.
Select the 'Resolve' option. This will create a line to add a resolving transaction. Enter a category to assign this to (eg Bank charges and fees or so on - if you don't see one relevant to you, you can create your own from this dropdown by selecting '+ Add New'.
Enter a date, a memo/class etc as desired, then enter the amount of the fee as a negative figure, then Save.
This will mark the invoice as fully paid and resolve the difference of the GST.