Please visit your QNB branch in person with your new offer letter & explain them the situation that your new work permit can only be done after you exit Doha & re-enter using the new visa. In the mean time keep funds in your account for at least 1-2 months EMI.

As mentioned above, the bank may ask a guarantee letter which your new employer can easily provide or any of your friend who has a descent salary & works with a listed employer in QNB's list of approved companies.