What is the ruling on a husband leaving his wife and children in a foreign country and traveling with his niece as a mahram on a flight lasting four days or less, and is there sin in refusing the brother's request?

A mahram is required for a woman's travel, but her stay in another country without one is permissible if there is no fear of fitna (temptation/strife) and no suspicion arises.

If you feel secure regarding your family, there is no sin upon you in traveling with your niece. However, if you do not feel secure, then it is not permissible for you to leave them alone.

You are not obligated to travel with her, and you will not incur sin by refusing to do so. However, it is preferable to fulfill your brother's request if no harm will result to you.