how does one person handle the religious requirements of 150+ member worlds?
If they are like modern military chaplains, Starfleet chaplains would be trained to serve all religious, faith and spiritual needs. No matter any individual or groups affiliation. And also those with none.
Starfleet chaplains can advise the ship's commanding officer on issues of religion, ethics, morale and morals.
Starfleet chaplains can meet with planetary religious leaders to understand the role of religion as a force for reconciliation and peace.