Is there a sin upon the questioner for not giving her father the money from the personal loan directly, after she gave it to her brother, the agent, to deliver it to the father, and he refused to do so?

What you have mentioned does not obligate you to pay anything from the loan or otherwise to your father; rather, it is purely a voluntary donation from you, and it should be given to him directly without an intermediary, as long as you doubt your brother. If you give the money to your brother for him to give it to your father, he is obligated to deliver it to him, and he is not allowed to take it for himself, for he is entrusted. Allah Almighty says: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]."