Is it permissible to sell dollars to a friend at their real price — not the one announced by banks — in order to avoid the loss incurred by exchanging them through the bank?

There is no harm in selling dollars at the agreed-upon price, provided there is immediate possession, based on the Prophet's (peace be upon him) saying: "If these types differ, then sell however you wish, if it is hand to hand." It is not necessary to sell dollars to the bank at a loss.