One option you could pursue, while trying to stick to canon, is have Kaji be the time traveler. However, rather than having him going back from 2015, as his past is not significantly detailed, what if he were written in as originally being born after Third Impact? He could, like "The Terminator", be the offspring of the survivors, who still find themselves trying to rebuild from all the damage of the Angel Wars, the attack on the GeoFront, and Instrumentality.
Now, imagine Unit-04 and Nevada suddenly reemerge after decades of vanishing in the Nevada desert, and it is discovered the overloaded prototype S2 engine's sabotage actually led to an accidental leap through time to this post-Instrumentality future. What if Kaji, as a last ditch effort, was chosen to go back, make contact with the significant players of the final days of man, and see if he could turn history onto a slightly different course?
So, Kaji, instead of being just a spy for the Japanese government, the U.N., and an employee of NERV, would in fact be in those positions deliberately to interfere enough with the plans for mankind to avoid Third Impact. Perhaps he also finds that Third Impact isn't really avoidable, but the aftermath is, to an extent, controllable.