My understanding is that Troi (at that time) was a staff officer in the medical section, in the chain of command she likely worked for Beverly Crusher. Either that or she headed her own section.
Star Trek sometime has strange ways of doing things, but in today's military Troi would be legally incapable of giving O'Brien a lawful order concerning ship operations.
And I think O'Brien knew this.
didn't want to separate the ship. He has been shown to be intensely loyal to his superior officers and his fellows. O'Brien basically bullshitted Ro and Troi into thinking that Troi outranked Ro on the bridge. Which Troi didn't. O'Brien reveals later in DS9 that he himself has been in command of ships, separating was the wrong option.
Ro really should have known she actually was in command from her time in the academy, but junior officers are told
to listen to senior NCO's.
Later Troi would qualify as a bridge officer, only then could she legally give command orders.