As per Season 6 Episode 8, "No One", the members of Brotherhood Without Banners which attacked and killed the villagers and the septon are shown to be a splinter group which did an unsanctioned attack.
The Brotherhood Without Banners' leader, Beric Dondarrion, Thoros of Myr and company were about to hang them (the splinter group) for murdering the villagers, until they met Sandor Clegane. Related dialog:
Thoros: Clegane. What the fuck you doing here?
Clegane: Chasing them. You?
Thoros: Hanging them.
Clegane: Any particular reason?
Beric: They're our men. Or they were.
Beric: They attacked a nearby sept and murdered the villagers.
Beric: Why do you want them?
Clegane: Same reason. I was helping build it.
Beric: It's the Brotherhood's good name they've dragged through the dirt.
This shows that the Brotherhood Without Banners have not turned to banditry, don't approve attacking innocents simply because they are of another faith, and they were not specifically targeting the septon or Sandor Clegane.
They also don't see Clegane as an enemy. They believe that Clegane winning the trial by combat against Beric is the Lord of Light's will. They even asked that Clegane join them as they make their move northwards:
Beric: You ought to join us. We could use you.
Clegane: I tried joining. Didn't work out for me.
Beric: Cold winds are rising in the North.
Clegane: And you're going to go stop them?
Beric: We need good men to help us.
Clegane: Last time you saw me, you wanted to execute me.
Beric: True enough. But the Lord of Light gave you the power to defeat