They could have legitimately considered it a high offense and raided the Sept of Baelor, demanding her return or threaten to kill all opposition until the "rightful ruler" is returned. They didn't do so. And attacking an armed force who is accused of wrongfully kidnapping an important figure while she believed to be in complete security under the laws of hospitality doesn't sound to me exactly like going against the religion itself, but rather simply against one of the administrative branches that has gone out of bounds. Anyways, I think that yes, they could have attacked the Sept and rescued her without the entire Faith turning against them. At least, until Cersei's reported crimes are publicized.
I tend to think that if Arya was meant to become a lesbian, then we'd have had some indications of that by now. Maybe as her being particularly tender towards girls, or something. I think most people are already aware of their preferences years before ever reaching puberty.