But I'm not sure Kira would have taken kindly to anyone, even her own CO, interfering in a personal religious matter.
Because that's exactly what orb experiences are. The Orbs are religious icons, and Starfleet is not in the business of restricting the personal religious freedoms of anyone (especially those who aren't in Starfleet).
And as I said, not even the DTI can take control of the Orb (it's property of the Bajoran government) or forbid anyone from using it. If the Bajoran government says you can use the Orb, then you can, and the Federation has no say in the matter.