A Minbari does not kill another Minbari. Simple as that. What can be worse and more massive as a violation of that sacred law than a civil war? How could a civil war happen with such a rule in place?
I assume you refer to the 2261 Civil War postdating Valen's edicts.
According to the wikia article they started out by denying resources (indirectly causing death) and ended with bombardments. With outright killing in between. So escalation overruled law.
By 2261 the peace between the Religious and Warrior castes finally broke. In some major cities and communities, the warriors no longer wished to lend aid and support to their Religious Caste neighbors. The warriors had apparently stolen from them and destroyed a great number of resources belonging to the Religious Caste, leaving whole populations of priests to fend for themselves. Some of those displaced were forced to journey to an adjoining population center to find food and supplies, and to meet with others of their brethren. Many of these refugees died of exposure before reaching such places.
The situation on Minbar soon turned to outright violence and war. It was because of this that Delenn decided to return home to organize her caste and possibly mediate the conflict. She met with Neroon, much to the chagrin of others in her caste, to resolve the situation peacefully for he has the ear of Shakiri - a prominent leader in the Warrior Caste.
Delenn soon convinced Neroon to help her end the war between the castes. At her urging Neroon apparently betrays the Religious Caste, and leaves to meet Shakiri - having acquired intelligence that would greatly benefit the Warrior Caste. The Warrior Caste then began its final assault on the strongholds of the Religious Caste. Betrayed and defeated, they huddled together as they were bombed into complete submission. Delenn then had Lennier send a message to the Warrior Caste announcing their surrender.
It ended with personal sacrifice. While surrendering Delenn invoked a ritual challenging the head of the Warrior caste, proving him unworthy and effectively ending all strife.
Delenn interrupted Shakiri to invoke the ancient tradition of sacrificial rites to prove the worthiness of leaders - as she and Neroon had earlier planned. She challenged Shakiri to step into the Starfire Wheel and die for his Caste. At first he refused but then chose to join her in the circle. Unable to stand the energy of the Wheel, Shakiri leaped out of the energy field - proving that he would not die for his caste and was unworthy. Despite having told Neroon that she would leave the energy field after Shakiri did, Delenn had chosen to stay in to prove herself and set an example for all Minbari. But Neroon, not wanting to see her die, and finally realizing that the calling of his heart was religious, not warrior, stepped into the Wheel. Picking up Delenn and handing her to Lennier, he stayed in the Wheel. He proclaimed the true calling of his heart to the Minbari, and implored the people to listen to Delenn. A few seconds later the Wheel reached its full and lethal energy, incinerating Neroon.
A few days later, still suffering from the effects of the Wheel, Delenn formed a new Grey Council.
Well, there's civil war and there's civil war. It doesn't have to be like our traditional human civil wars. I see several options...
Minbari don't kill Minbari; bullets kill Minbari
In other words, they can use word-definition sophistries to de-Minbarize the opposition, or what their own actions mean. "Hey, I just pushed an anvil off the ledge; how was I to know those dopey rebels would fail to step out of the way?" Could be it's okay to hire non-Minbari mercenaries to do the dirty work... ;D
Business is war
Could be the civil war is run more like a hostile corporate takeover, the idea being to pauperize the opposition. The losers of the economic battle will lose so much face that they'll lose their political status as well.
It's such a pain to time your kinetic strikes on enemy property such that the building is completely empty, but it is sure satisfying!
Hey, if Shadows and Vorlons can manipulate stooges into fighting each other, why not Minbari? The Min factions can pick proxies to do their fighting; the side whose proxy wins, wins the civil war.
Veggies are alive, yes?
Some Minbari are telepaths. Who knows what kind of awfulness they can do to each other while leaving the victim "alive"... ;D
Simple. Proclaim people on the opposing side (which are Minbari by species) not to be TRUE Minbari™. If they are not TRUE Minbari™ , no protections in law work for them. In fact, helping them would make you be NOT TRUE Minbari™ as well. That means no contracts, no common decency, no protection against killing! You do not help a rabid dog, you put it down. You do not negotiate with NOT TRUE Minbary™, you do not help them, you put them down, rabid traitors that they are!