I never imagine Morag Tong members using poison to kill. It just seems beneath them to me. My opinion though. In Morrowind, I walked up to the target, let them know what was to be and ended their life, then flashed my writ to any guard dumb enough to interfere.
Just play the type of assassin you want with the skills you'd enjoy using. You don't have to follow the norm. A idea though, you could say you have one last writ to perform despite the current situation of Morrowind and the Morag Tong. One last target and you have vowed to find them and end their life.