To avoid negative connotations, show a positive one.
A really good example here is Judge Frollo from the Hunchback of Notre Dame (as the negative connotation)
Note that I am intentionally choosing the Disney movie, not the book, because Disney spent a lot of time making sure the story was palatable to a largely Christian audience. The book, however, took a reasonably strong (for the time) anti-religion stance, so it's not a good example here.
“Judge Claude Frollo longed to purge the world of vice and sin.
And he saw corruption everywhere, except within.” ―Clopin
Frollo is a villain who is driven by religion, or at least his interpretation on how religion should be implemented. He uses it to oppress both others and himself, and he rules through fear and violence. He spent years (at least from Quasimodo's birth to the present day) persecuting gypsies, effectively trying to ethnically cleanse Paris.
In pursuit of his persecution, he has little regard for collateral damage.
And he does it in the name of God. It is his justification for everything. And the movie never actively dispels his justification. There is no literal word of God which disavows Frollo.
Also note that Frollo is genuinely religious. The reason he doesn't go through with killing baby Quasimodo is because he fears divine judgment for his actions. If he were only faking his belief, he would likely not have kept Quasimodo alive.
However, as these past events are a story being told by Clopin to children, it's possible that this is not what actually happened. We can't know that for certain, but there is no indication that any of Clopin's story is untrue. He seems to be a knowing narrator, one whose character is also tangentially woven into the plot.
It's very easy to interpret this move as being anti-Christian. But this movie isn't anti-Christian. Why?
Because of the archdeacon
Notice that black vs white contrast. It really just boils down to how they present themselves in terms of Christianity.
Note: in the book, Frollo is the archdeacon, but not in the movie. This is presumably part of the same move to remove the connotation of the Church being evil.
“See there the innocent blood you have spilt on the steps of Notre Dame.
Now you would add this child's blood to your guilt on the steps of Notre Dame.
You can lie to yourself and your minions,
You can claim that you haven't a qualm.
But you never can, run from nor hide what you've done from the eyes, the very eyes of Notre Dame”
―The archdeacon to Frollo after murdering Quasimodo's mother and attempting to murder baby Quasimodo
Here is the link to the archdeacon's first scene. It's his most important scene and sets the stage for his character and how he is not like Frollo, right form the outset of the movie. The video explains it better than I can with words.
The archdeacon dispells the viewer's possible interpretation that the Christian faith is the real villain here, by showing us a good Christian.
He rescues Quasimodo from being dropped in a well, he provides sanctuary to Esmeralda at a time where gypsies are actively being hunted and persecuted, tries to intervene in Esmeralda's execution, and he is the only character who has consistently pointed out Frollo's evil deeds with little regard for how doing this would impact him (by getting on Frollo's bad side).
When you consider that Frollo is the archdeacon's superior in the movie, this proves the archdeacon to be a stalwart beacon of good and kindness.
This saves the image of the Catholic faith, as it ensures that it is not only shown in a negative light. It is made apparent that Frollo's justifications are fabrications for personal gain (he is a fascist at heart, and uses God as his excuse to enact his own will), whereas the archdeacon is selfless, kind, and stands up for what is right.
The conclusion the viewer now draws, instead of assuming that evil Frollo represents the (therefore presumed to be evil) Church, is that evil Frollo uses the good Church as a guise for his villainy.
Does your story think religion is inherently evil?
This group is intended to be the antagonists and would be illustrated as irredeemably evil and corrupt, largely because of their devotion to their faith.
The crux of the issue here is the same as for Frollo. Are they evil/corrupt because of their faith, or because of an overzealous devotion to a faith that (by itself) is not corrupt?
If you take the stance that the religion is inherently corrupt down to the core, which is perfectly fine (and what Victor Hugo intended with Frollo in the book), then you should either accept that those who disagree with your stance will not like your book, or you should use an alternate fictional faith to reduce (but you won't completely avoid!) the amount of offense that will be taken.
If your stance is that this group is corrupt, but not necessarily the faith in and of itself, show the difference. Show some religious people who are not corrupt. Show some corrupt people who are not religious. Dispel the idea that corruption and religion go hand in hand, and make it clear that your plot villains are both corrupt and religious, but that these are two unrelated traits.