How can the issue of impermissibility of embracing non-Muslim female cousins be explained to them, knowing that this custom is prevalent in their societies, without alienating them from Islam?

What the husband did—hugging his adult female cousins—is a sin that cannot be justified by the claim of inviting people to Islam and not alienating them. The caller to Islam is the first person who should be steadfast in adhering to Allah's law. The end does not justify the forbidden means, and there is nothing in Islam that one should be shy to apply. The husband must clarify the rulings of the Sharia (definition) to his family and relatives, and both spouses must adhere to Allah's law in what they do and say, and beware of neglecting an obligation or committing a forbidden act for the sake of inviting people.