There are some good ideas here. Hopefully, I can add a couple more.
The Black Templars are led into battle by their Chaplains. While their faith is nigh-unshakable, it's not flawless. Witnessing a particularly horrendous act by a governor of an Imperial world against loyal citizen, and not being able to prosecute the governor for whatever reason (protection by the Adeptus Ministorum, an Inquisitor, etc.) could make them begin to question their faith. Eventually, that can lead down the road to the realization that the Imperium is totally corrupt, and his sermons begin to alter accordingly.
This is sort of an Iron Warriors angle, as far as how they fall. Space Marines take their recruiting very seriously, and while most recruit from more than one world, the Black Templars recruit from dozens. However, some worlds are more steady suppliers of acceptable recruits than others. Perhaps a particularly important recruiting ground to the Black Templars has been found guilty of treason/heresy/tax evasion/whatever on false evidence. A crusade is launched and is successfully prosecuted. The Marshal of the crusade discovers, in the final moments of the crusade, that he's not only exterminated millions of loyal Imperial citizens, but he's destroyed one of the most valuable recruiting grounds the Black Templars have. He takes it as an affront to the Chapter and to his honor, as do many of the Crusaders. They then lash out at the corrupt Imperium that forced them into such a situation.
While psychic screening is usually pretty flawless, particularly when Space Marines are involved, it doesn't mean that latent abilities can surface much later. What if a Marshal or Chaplain develops psychic abilities, and becomes possessed?