Is it permissible to buy a car from one's brother through an Islamic bank, such that it is formally registered in the buyer's name but remains the brother's property, while the buyer benefits from the car's price due to his urgent need for money?

It is not permissible to nominally purchase your brother's car through an Islamic bank for the purpose of registering it in your name while it effectively remains his, because this is a circumvention to obtain an interest-based loan. This is what Ibn Abbas, may Allah be pleased with them both, described as "dirhams for dirhams with an excess, with a piece of silk entering between them." There is no difference between this and exchanging one hundred for one hundred and twenty without trickery. If you are in need of money, you may resort to the permissible Tawarruq.