From Lindelof's interview it sounds like they just changed the dialog for Cumberbatch's character.
How did the script change, if at all, when Benedict Cumberbatch signed on?
Lindelof: Well, it changed in terms of anytime you cast any actor, even though you think that Kirk is a fixed thing and you’re not going to change him at all, when we cast Chris we rewrote the part to basically match up with what we viewed Chris was doing. Because these parts are not off the rack suits where you put them on and they fit like a glove. You make the suit and then the actor puts it on and then you say, “I’ve got to now tailor this thing so it fits them perfectly.” So Benedict was no different. The kind of actor that he is-
Which is spectacular.
Lindelof: Which is spectacular, necessitated a certain shift in just the way that the character was going to sound, you know? Our own inner ear for that character we were like, “O.K. we wrote this character John Harrison and this is what he said and this is what he did, but now we’ve got Benedict Cumberbatch playing him so let’s rewrite the movie with that in mind.” And that didn’t mean that John Harrison did anything differently, or it didn’t change the story in anyway, but it did change the words coming out of his mouth.
And the character was always John Harrison even before Cumberbatch was cast. Of course Harrison may not be his true name but a pseudonym for another well known character.
I don't think the Bad Robot guys care about getting the ethnicity all that right if JH is really Khan. They're more into the best actor for the role kind of thing. The could drop in a line about his appearance being altered to hide his true ID.