New Zealand Law
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I am sorry I answer questions as they come in, and so I have been a bit delayed by others ahead of you.
You would have a good claim under the Consumer Guarantees Act for a refund or replacement as the car is clearly not fit for purpose. A major defect like this should not occur so soon after the purchase, and I expect the dealer is just thinking about his liability under the act. But you are in a good position to insist on a refund.
Then the dealer who has expertise in checking cars may have to accept the trade in as it may be a caveat emptor situation. You showed him the trade in and he offered you money for this as a credit. If he didn't find the problem then it is his responsibility
He must repay the loan as that is part of the transaction. But certainly you are right to be concerned. I suggest you may want to consider getting the trade in independently examined in case this is not genuine
If he accepted that car then it is his problem. He has the expertise to look at cars and see if they are in good condition.