The problem with the Khan character in STID has little to do with race or being a Sikh or not. The problem is that, other than the name and the genetic engineering aspect, there is nothing recognizable about Khan in Benedict Cumberbatch's performance - or the way he was written.
Montalban played a handsome character who put on a charming performance as Khan, even if he was a dangerous psychopath. Cumberbatch's Khan is just a dangerous psychopath. With all due respect, he's not nearly as handsome and he's nearly not charming.
It's like he was cast as some different character altogether, then, right before shooting began, somebody handed him a new script that had Khan in it and he didn't have the time or inclination to actually change his performance to suit that character.
Of course, one can say that the character was changed in this reality beyond recognition. And if you're good with that, so be it. But for me, it makes it difficult to suspend my disbelief whatsoever.
Montalban's Khan is right out of the deep freeze.
Real world thought experiment: Adolf Hitler is woken up. Would he look the same and be highly recognizable? If you saw him casually might you say "holy mackeral, that's..."?
This society can make a human appear as a Romulan. Plastic surgery or the equivalent is routine. Ferengis talk English due to invisible technology. The FIRST thing i would do, if i wanted an instantly recognizable face to be useful, is alter it beyond recognition. The second thing would be to change the voice tonality.
OK, your mileage may vary, but Marcus had a year so turn "Khan" into Harrison and i think altering his appearance, drastically, would be expected.