I totally understand your frustration!We are in the same boat as you in that we want to be done by Christmas..
Don't understand why the bank needs to know that you are paying for the variations?
For our variations outside of contract variations, we had to pay Wisdom upfront before construction began. But if we wanted them included as part of the progress payments, we had to make that arrangement with the bank. The bank would then need to see you have the money to cover that bit not part of the construction loan amount. However that was the arrangement with our bank, other banks might have other arrangements.
At the end of the day, Wisdom need to see you have money to pay for the build. If your are borrowing from the bank, then a letter from the bank re finance approval + payment of money for variation outside of contract. If you have paid for your variations from another source, I guess the bank just wants to make sure they have been paid or will be paid, before construction begins. They probably want to make sure the house will be built (and not halted due to lack of funds from a second source) and paid for in full, as they have security over the house to cover the construction loan.
Hope you get it sorted soon. Hard to understand something so simple as paperwork can take so long to get sorted.