What is the ruling on benefiting from money received from selling a car via Murabaha to an Islamic bank, if the buyer had already taken possession of the car before the procedures were completed and the contract was signed between the seller and the bank?

The aforementioned transaction is valid because it fulfills the pillars and conditions of a Murabaha sale to a purchasing applicant. There is no impediment to the buyer following up on the procedures, and this is considered an authorization from the bank.

The most important aspect of a Murabaha sale is that the bank purchases the commodity and then sells it to the purchasing applicant. Since the questioner signed the sale contract between himself and the bank, this means that the bank purchased the car and then sold it to him, thus the transaction is valid.

As for the buyer's request to increase the price in order to gain an increment, it is a correct action on the part of the questioner to refuse to comply, because the buyer, through this falsehood, reaches what is unlawful and takes the increment to return it with interest, which is usury (riba). Aiding in it is religiously forbidden due to the lying and deception involved towards the bank.