I'd go even more simple, and take the initiative order and damage out of the equation altogether.
Maybe something like this?
Assassinate
If the assassin scores a critical hit against a living creature, he can choose to have the target make a Constitution save throw (DC = 8 + proficiency bonus + Dexterity modifier). If the creature fails this save throw, its hit points are reduced to 0. Large or larger creatures make this save throw with Advantage. The assassin must finish a long rest before they can use this ability again.