Is it unlawful for me to inform an Islamic bank of a price lower than the car's actual price, knowing that the bank calculates its commission on the full amount, and if so, what should I do?

This action is impermissible due to the presence of lying and selling what one does not own. If you have paid an amount to the seller, the remaining price becomes a debt owed by you, and selling the car to the bank after that is an invalid sale. The legitimate solution is to inform the bank of what has transpired and request the annulment of the sale. Then, the amount paid to the seller should be reclaimed, and the bank should buy it anew from the seller and then sell it to you.