I think 'John Harrison' may be either a pseudonym of "John Arrik Soong", son or grandson of Arrik Soong, or that he may be an alternate-universe-accelerated Soong-type android, sort of a re-imagined Lore.
We know that Noonian Soong was able to create androids which both had emotions and appeared perfectly human ('Inheritance') and while that's a lot of canon to be throwing at the mass audience the film is courting, that same mass audience will likely just happily accept a human-appearing android without needing to be given a justification for it.
Arrik Soong's knowledge of genetic engineering/gene therapy would account for Harrison's claim to be able to help the little girl. His being an android would plausibly account for his apparently superhuman abilities and nerve-pinch immunity.
Soong's treatment at the hands of Starfleet could also be the underlying vengeance motive.
The biggest weakness of this theory is his status as a Starfleet officer. Granted, there's Data, but I always got the impression that Data was somewhat famous within Starfleet just for being an android officer, so assuming that would hold true in this case, it would be pretty tough to clandestinely plan a bunch of terror attacks.
So perhaps Harrison is an augment, but of the Soong variety rather than a Khan descendant or analogue (or Khan himself).