Is refraining from hosting the paternal cousin who did not offer condolences on the death of one's children considered a sharia transgression, especially after his justification that he shared his feelings inwardly and that the presence of his mother and two sisters was sufficient?

Maintaining kinship ties is obligatory, and severing them is forbidden. The one who maintains ties is not merely he who reciprocates connection, but rather he who connects with those who cut him off, and treats well those who mistreat him. Do not fall short in maintaining ties with your cousin (son of your paternal aunt) under the pretext that he failed to offer you condolences, for the right of a Muslim is not contingent upon reciprocation. He ought to have offered you condolences if he was able. We advise you to overlook the past, forgive, and not revert to such behavior in the future.