What is the ruling on working on an investment project that relies on settling clients' defaulted loans by purchasing a car for them for a certain amount, then having the client write a check for a larger amount, while enabling him to sell the car and use the money to pay off his old loan and obtain a new loan, with the institution's profit being the difference between the two amounts?

There is no objection to the aforementioned transaction, provided that what transpires between you and the purchasing agent is merely a promise to purchase, not a sales contract, and receiving a down payment as a guarantee of seriousness does not affect it. Once you have purchased the car and it has entered your possession, you then finalize the sales contract with the purchasing agent through a murabaha sale. There is no objection to taking a document that proves the debt. If the buyer takes possession of the commodity, he is permitted to sell it to benefit from its price (tawarruq), and there is no harm in selling it to the showroom without a prior agreement to do so.