Am I obligated to offer an expiatory offering (kaffarat yamin) after swearing not to buy a gift for myself and my daughter from the 'Umrah, then accepting a gift for my daughter that my brother bought with a prior intention for her success?

It appears you have broken your oath and are required to offer expiation, unless your intention was to specify the gift for your brother's arrival. In that case, the gift would be for your daughter for her success, and no expiation would be necessary for you. If you did break your oath, its expiation is to feed ten poor people, or clothe them, or free a slave. If you cannot find any of these, then you must fast for three days. We advise you to accept the gift and offer the expiation, as it involves mending hearts. The Prophet, peace and blessings be upon him, said: "Whoever takes an oath and then sees something else as better than it, let him do that which is better and expiate for his oath."