It should be child's play for the Director to argue that Kagari and Chou killed each other. Evidence can be falsified to make it appear so. The likelihood of anybody questioning it would seem slim since it makes sense that two guys could kill each other in a close-quarters firefight.
If the Director is particularly machiavellian, she might wax poetic about Kagari's great and noble sacrifice in defense of the Sybil system.
(of course, such statements could be exactly what sets off alarms for some people
)
But if the Director does this, I think that Kogami will find some small but crucial hole in "the official story". And that might reveal something very troubling to him. And that's when Kogami might go all rebel (perhaps even freeing Makishima from prison to do so, presuming Makishima isn't just executed). This could all lead to a critical decision by Akane.