Leaving aside any real world analogies, historically in Thedas there was an Inquisition. It pre-dated the Chantry and comprised a bunch of Andrastrian hardliners who tried to bring order from the chaos that existed in the period after her death. At that time there were numerous cults, some devoted to Andraste, some to something else. They were also concerned to ensure that the mages didn't establish a powerbase for themselves. This was the case, even in Tevinter, where for a time they maintained a front of non-mages in power, while the mages themselves actually pulled the strings behind the scenes.
Then along came the Chantry. They unified the various cults into one accepted version. They seem to have offered some degree of protection to mages provided they didn't use their power in any significant way. The mages objected to this and so a compromise was agreed whereby the mages would live outside the community but be able to practice and develop their magic. The Chantry then brought in the Inquisition and presumably suggested they police the mages to ensure things never got out of hand. So the Inquisition became the Templar order and the Circles were started.
Subsequently the Chantry split, so the Tevinter Chantry allowed mages back into government and actually had a mage as the Black Divine, whereas the Orlesian Chantry retained its stance on mages in positions of power. The Templars continued to operate in both places but have rather less power in Tevinter.
So the Templars have always been semi-autonomous but answer to the Chantry when it comes to major decisions, like calling an Annulment of a Circle. The White Chantry no longer have a military force to enforce their decisions but that makes no difference to the ordinary people of Thedas who continue to worship as they have done for the last 900 odd years. All the latest events mean is that they Chantry no longer controls the activities of mages or Templars. Whether having Templars come in a kill peasants they mistakenly think are mages will impact on the Chantry will, I suppose, depend on how aware they are of the current situation. I would imagine that the local Chantries will opt for damage limitation and would have already informed their flock of the Templars defection shortly after it happened. Cullen hinted that ordinary folk were already regarding Templars in a less favourable light than they once had, so most people will just see them as rebelling against their religious leaders and withdraw their support. Apparently the latest Inquisition was the idea of the Divine to deal with the situation, thus restoring a measure of military and political power to the Chantry but owing to her death it has become something of an independent organisation.
So from what we have been told, if you wish to play up the religious aspects of the Inquisition and your own place in it, you are free to do so. You can even have Chantry buildings in your keeps. However, if you don't wish to do so, that is also an option. Neither is likely to impact on the Chantry religion as a whole unless you choose to let the world be torn apart, in which case it won't really matter what people believe.