How can one advise a muezzin and a mosque caretaker who have personal disagreements over worldly matters, while they both maintain their prayers and service to the mosque?

You can remind them of the virtue of forgiveness, pardon, and accepting excuses, and that the one who initiates reconciliation and peace is superior. Also, remind each of them of the virtues and deeds of the other, and the necessity of interpreting their companion's actions in the best possible light, while avoiding ill assumptions.