Okay, I've got one of those theories about how you can kind of combine Khan and Mitchell into this storyline:
Remember, the timeline was altered by Nero's attack in a way that made the Federation ever so slightly more aggresive and advanced. Here, Gary Mitchell is a cadet fresh out of Starfleet who goes off on an exploratory mission into deep space, while there, Mitchell encounters the Botany Bay, which as per Starfleet's new policy, he immediately begins experimenting and researching on.
Mitchell comes to realize they have stumbled upon the Augmented warriors of Khan Noonien Singh, and he and the research team begin trying to weaponize the genetic enhancements. As the changes take hold, the crew rapidly begins to turn treacherous, especially Mitchell, leading the commanding officer to blasting the research station their on into dust.
However, Mitchell clings tenaciously to life, and their experiments have awakened the slumbering crew of the Botany Bay, at which point Khan recovers Mitchell and, impressed by his ingenuity and raw hatred, helps heal him. Mitchell is now a slightly lesser Augment than Khan and his followers, having an obsessive impulse to avenge his "betrayal" at the hands of Starfleet and willing to play a part as Khan's vanguard scout and prophet, with the goal of crippling the Federation and bringing the Klingon Empire to heel underneath the coming of Khan.