How to maximise the odds of becoming Cathar?

If you really want to become Cathar, I think there are more effective ways of doing this. I can think of two:

• Have your successor educated by a Cathar character (if they are diligent and zealous it will speed up the conversion)
• Invite diligent and zealous Cathar characters to your court - they have an event that can convert random courtiers to their religion (you can then choose to convert)

Also, note that heresy in general spreads faster if moral authority of your religion is low. You can help it along by appointing an anti-pope.

If you really want to stick with the court chaplain research, you can see the chances of his "failure" in the tooltip - failure means he becomes a heretic. I think the game just picks a heresy at random though, so I think this could take a long time.