Future perfect/simple future/present perfect

Please consider the following: They will report to me what they will have known/will know/have known. Which one is right if the context is they will report to me after they know it

The clause what they know is a relative clause. In integrated relative clauses, a simple present can be used for a future situation (CaGEL p135).

The natural interpretation, then, of:

They will report to me what they know

would be that whatever they report to me, they will know when reporting it (Which is quite obvious if we think about it).

All of the other choices you've listed will know / have known / will have known would not sound natural and give a strange interpretation.

will know - This would imply that they'll report of the things they will find out about after the future instance of reporting you are referring to.

have known - this would imply that they found out some piece of information and subsequently forgot it before reporting it, but do remember that they did know it.

will have known - same problem

In fact, I'd just replace the the what with a when and be done with it:

They will report to me when they know