Is what the questioner did—not sending photos of her brother's newborn to her other brother and refraining from informing him that she was made to swear an oath to that effect—considered hypocrisy, incitement, or disobedience, especially since her father ordered her to lie?

If your brother prevents you from sending pictures of his new baby, a general prohibition that also includes your other brother, then your apology to your mother and your other brother is not a lie. Your disobedience to your mother's command to send the pictures is not considered forbidden disobedience ( Uquq); because fulfilling an oath is recommended (mustahabb), and abandoning a recommended act is not a sin. The pictures are a trust with you, and sending them would be a betrayal. The forbidden disobedience (Uquq) is disobeying parents in matters that are not sinful, as the Prophet, peace and blessings be upon him, said: "There is no obedience in disobedience to Allah; obedience is only in what is right." Informing your other brother of the truth was not advisable due to the potential strain it could cause in their relationship.