Is it permissible to agree with the seller to increase the price of the apartment for the purpose of obtaining additional financing from the bank to cover furnishing costs, knowing that the bank is the one who suggested it?

You are not permitted to do that, as it is a trick to get around usury (riba). The bank will take interest on the excess amount, which is not genuinely part of the price, making it a pure usurious loan. There is no difference whether the bank knows about the trick or not; in fact, its knowledge of it and approval of it is worse.