Is it obligatory to perform a supererogatory prayer with the congregation of the mosque who pray Fajr half an hour before its time, then return to pray Fajr on time as an obligatory prayer, based on the hadith "Pray the prayer at its appointed time, and if you find them praying, then pray with them, for it will be a supererogatory prayer for you," or is it sufficient to pray Fajr at home to avoid hardship?

You are not required to pray in the mosque if the people of the mosque pray Fajr before its prescribed time, nor is that demanded of you. The hadith (definition) of Abu Dharr pertains to delaying the prayer beyond its preferred time, not to performing it before its time. The meaning of "they cause the prayer to die" (يميتون الصلاة) is that they delay it beyond its preferred time, not beyond its entire time. The command to repeat the prayer with the congregation in this hadith is for recommendation, not for obligation. It is recommended for the worshipper to pray it individually at the beginning of its time, then to pray it with the Imam to combine the virtues of praying at the beginning of the time and praying in congregation.