So what is the verdict ?

The two suspects (the Klingon and the human) are separated by a space-like distance (s2<0 ).
For space-like distances, the order of events is relative
But events separated by a space-like distance do not influence each other.
When the Klingon and the human were firing, they were not aware that the other is firing, too. Actually, they only see the other firing when the victim is already hit by their own laser beam.

The two suspects (the Klingon and the human) are separated by a space-like distance (s2<0 ).