Is it permissible to congratulate the wife's sister, who lives in a European country and whose daughter has given birth to a child from a Christian husband, knowing that not congratulating her might cause her anger due to the wife's family's attention to such details?

First, the sister must be rebuked, and she must be reminded of the gravity of her marriage to a disbeliever, which is unanimously prohibited. If she does not benefit from the advice, her family should shun her if shunning would benefit her, because the religious benefit is what matters. As for congratulating her on the newborn, that is not appropriate, because it implies an endorsement of this reprehensible act and might encourage her to persist in her error. There should be no compromise in this matter, for Allah's pleasure comes first. Whoever seeks Allah's pleasure despite people's displeasure, Allah will spare him the burden of people. And whoever seeks people's pleasure despite Allah's displeasure, Allah will entrust him to people.