What is better for you when going to the Haram: praying Maghrib at home immediately after the adhan, or in a mosque along the way as a second congregation, to ensure securing a spot in the Haram? And must I refrain from going with my father to the Haram if he refuses to pray in the mosque as a first congregation?

Congregational prayer in the mosque is obligatory for able-bodied men, and abandoning it without an excuse is a sin. When mentioned generally, it refers to the congregational prayer in the mosque, provided there are people and the Muslim is able to perform it with them. What you mentioned is not an excuse for missing the congregational prayer. You can break your fast in the mosque and then go to the Haram. If your father insists on performing the Maghrib prayer at home, you may agree with him and obey him, as most scholars see no harm in performing congregational prayer at home.