I don't want to hear about sports. You may not want to hear about religion. We don't have a right to not hear those things we don't like.
an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech,
That means exactly what it says, BeHoldAPaleHorse. No matter how much you (and the ACLU) want it to mean something else. Neither you personally, nor your group of hypothetical nonreligious troops have a right to abridge the chaplain's freedom of speech. Nor can the chain of command lawfully do so.
Do you have the right to not be forced to hear religious oratory that is not of your faith? Or does your local preacher have the right to have the sheriff drag you into his church?
You see, attendance at military formations is mandatory. If you do not show up, you can do time in the stockade--the military equivalent of a jail. If you attempt to escape the stockade, you can be shot.
So, essentially, attendance at this chaplain's invocation is mandatory for servicemembers, on ultimate pain of death.
So, at such a formation, suppose a Roman Catholic chaplain starts praying the rosary, which many Protestants view as idolatry or pagan goddess worship.
Do the sailor's rights to worship as they see fit enter into the equation at all? Or is the chaplain's right to spout off senior to those sailor's rights, or even to the commanding officer's authority?