What should a young Syrian man working in Lebanon do, who fears carrying religious content on his mobile phone and wishes to work in a mosque affiliated with Dar Al-Fatwa, while some shaykhs forbid him from doing so, claiming they are innovators?

If the institution belongs to Ahl as-Sunnah, then there is no harm in working as a mu'adhdhin there. This work is better than working in Christian-majority areas where it is difficult to openly practice the rituals of the religion. It is permissible to work in Christian-majority areas if the work is lawful and there is no temptation (fitnah) regarding one's religion; otherwise, one must leave and work in Muslim-majority areas. Whoever strives to seek what is lawful and protect their religion, Allah will assist them and grant them success.