Hitlawn had dispatched four assassins in total to kill the escapees of the Lan Clan. The one that put up this mission was the Lan Clan itself. After all, they couldn't kill their own family members, especially those hailing from the main family line.
If word of it got out, it would be a blow to their reputation. And hence, they commissioned Hitlawn. And in truth, the major Clans were the biggest customers of Hitlawn, a reason this criminal organisation could exist to date.
In tricky situations like this, the task would be given to Hitlawn. Neither the reputation nor the consequences of these actions would affect the Clans that put up the mission. All infamy would be borne by Hitlawn.
So, it was somewhat of a win-win situation for both.
To the Clans, the high price required for the assassination wasn't a problem, in contrast to the perceivable blow to their reputation if they killed one of their own.