Is refusing a mother's request to drive her somewhere an act of disobedience, given that the car does not belong to the questioner and its owner does not permit its use?

If your brother has not given permission for you to use his car, then you are not entitled to use it. Your refusal to obey your mother in this instance is not an act of disobedience (cukuq), as obedience is only required in what is good and permissible (ma'ruf). You are obligated to obey your mother by transporting her to the desired location through any means you are able to provide.