When you trade a car in though, they just take the amount of the trade off the purchase of your new car, so really your new loan with the car yard would be the price of your new car minus the trade value, I don't think they just hand over cash with a trade. So your bank is saying that you wil still have the debt with them, but no car to hold against that debt as the car yard will have your new car as equity against your new loan.
Can you not talk to your bank about extending the loan you have to cover the extra cost of the new car, rather than using the car yard finance?