You don't. I'll bet you anything that they've priced it that far below invoice by using all the available dealer cash, and dealer cash works with VCI standard APR, 3rd-party financing, a duffel bag filled with cash, etc, just not with their lowered APR programs (Special APR or Dealer Choice) or special lease rates. With the car already in transit, tell them you want that price, but you want to finance through your credit unions, whose rates are almost certainly lower than VCI Std. If the dealer's finance office has an arrangement with your CU (they probably do), tell them you'll let them submit the loan application and collect their fee (which costs you nothing), rather than applying for the loan yourself.
EDIT: There is no way that price is contingent on using VCI Std rates. I just looked this up for someone else a few days ago; the current cash incentives on Alltrack are $3000 dealer cash plus $1000 Summer Sell-down bonus cash. Add those two numbers up and then compare that figure to the difference between the dealer invoice price and what you were quoted, and you can't help but notice the similarity. VW Dealer cash incentives can be used with 3rd-party (ie bank/CU) financing, and they know it. Get on the VWPP website, find out what the partner price is, and then subtract the $4000. If that number is lower than what they've quoted, that's what you should pay, and you can finance it through whoever you'd like.