How is it possible to visit peaceful disbelieving relatives and maintain ties with them without showing them familiarity or smiling at them, taking into consideration the prohibition of sitting with transgressors in a familiar manner, and that smiling at them, showing cheerfulness, familiarity, laughing with them, and showing them respect are among the prohibited forms of loyalty to disbelievers?

The statement mentioned in the question is not absolute; rather, it involves details and restrictions: Sitting with transgressors while they are committing forbidden acts without enjoining them from doing so is unlawful. This may even reach the level of major sins if one who is able to forbid does not do so, or if one who is unable to remove the evil does not leave their company. As for smiling at a peaceful disbeliever for a valid reason, such as to incline his heart towards Islam or due to kinship, it is permissible. However, one must beware of following him or affirming him in his falsehood. The boycotting of a transgressor who persists in his transgression has been explained in a previous fatwa (definition).