Ok I know this is a stretch but..
We know that PJ and BK have shaken hands, so I does tick that box, he is also the right age and physical build. I personally think that it would make sense to be him just because it would be unexpected because he is supposedly dead (just like Partridge).

This is only my personal opinion but I think that the scene with him a PJ when he is trying to get names from PJ is very significant and may be a clue. He tells PJ that he wants to find RJ because he killed his brother or was responsible for his death. What if BK is RJ and he in-fact rather than actually killing his brother he did not stop him from being killed, e.g. did not help him in a time of need during his childhood.

This could lead to the hidden phobia that RJ has, that is the fear of failure or Atychiphobia. This would explain 2 things, his internal conflict about his failure in saving his brother and also his reaction to PJ criticisms that lead him to kill his wife and child. As well as this it would also explain why the counselor questioned his ability to handle hardship or tragedy, seeing that it had actually shaped him as a person. His Atychiphobia would also explain why he is so careful and never leaves any clues behind.

Lastly the fact that he broke in to PJ office to steal the list of RJ names could have simply been his way of 1. clearing his name as RJ, showing that he was interested in catching him and 2. making sure the PJ didn't have anything that could lead to him being found out as being RJ. The scene where he cuts off the mans finger for information could be another point of proving his innocence to PJ.

An interesting twist to this would be if no one actually knew he was RJ even the members of tyger tyger. This would further remove him from suspicion. His death could as well have been faked, even Rigsby comments on the fact that there is very little follow up and even the report looks dodgy.

I know this is a stretch but it would be an interesting twist.
I could imagine Patrick walking into RJs house and seeing childhood photos of him and his brother and then seeing how it caused him to kill out of his own internal anger.