The Klingons wouldn't much care who Dak'Rah ordered killed in combat. They were put out over (believing) he ordered his own men killed. But if they believed he fled in cowardice, he would be deemed to be without honor - utterly reprehensible and barely even worth the dignity of a painful execution. His family (I assume a general would have a big household) would be dishonored for generations. Dak'Rah, being entirely mentally deranged (from a Klingon psychological perspective), was committed to never, ever allowing the truth to come out.
So once it became clear M'Benga knew the truth, one of them was going to have to die. So the options I see are:
- Dak'Rah tried to kill M'Benga and wanted M'Benga dead (M'Benga justified).
- Dak'Rah started a fight, willing to kill M'Benga, but realistically knowing M'Benga probably could kill him (suicide by M'Benga).
- Dak'Rah killed himself (M'Benga innocent).
What I don't think happened was M'Benga deciding to "finish the job" and kill Dak'Rah.