Does shunning a cousin mean eternal damnation if it was due to his inappropriate behavior towards the father, given that no reply was received to the initiative for reconciliation?

As long as you greet each other upon meeting, there is no unlawful abandonment according to the majority of scholars. Some scholars, however, are of the view that abandonment does not cease until the relationship returns to its previous state. Al-Hafiz Ibn Hajar held that avoidance ceases simply by giving and returning greetings. As for Imam Ahmad and Ibn al-Qasim, they believed that avoidance is not absolved until the situation returns to what it was initially, or if the cessation of communication is hurtful, then avoidance is not broken by mere greetings. You have done well in sending a condolence message. As for the threat of eternal damnation in Hell for one who refuses reconciliation, it is understood in two ways: First: one who deems severance permissible without cause, knowing of its prohibition—such a person is a disbeliever who will abide eternally in Hell. Second: that he will not enter Paradise with the foremost, but will be delayed as a punishment for him. And whoever does not believe in the permissibility of severance, or believes in it while being ignorant of its legal ruling, does not commit an act of disbelief.