Is it obligatory for someone who suspects they have apostatized due to jokingly telling their younger brother that women sometimes do not pray, then immediately clarifying that they were joking, to make the intention to re-enter Islam?

We advise you to stay away from obsessive doubts concerning apostasy. Doubting apostasy is of no consequence, because he whose Islam is established with certainty does not exit it except with certainty. What you have mentioned does not indicate that you have committed apostasy, but you must repent from lying.