Not if the technology had been outlawed and suppressed - which is exactly what happened after the Eugenics Wars
Genetic augmentation of humans, yes. Not the procedure to create whatever heals cells damaged by radiation. You don't have to clone an entire human to do that. Kirk is not an Augment after getting Cumberkhan's blood.
It's possible that the healing factor wasn't something Khan always had, but something that he developed as part of his work with Section 31 after his revival. Maybe 23rd-century tech was able to activate a potential in Khan's biology that he didn't know was there.
But we don't know how it's done. It may be that the only way it works is by modifying a host body to create this magic blood. Henry Archer died a painful death from Clarke's Disease and according to Arik Soong he could have been easily saved if not for Earth's ban on genetic engineering. Right or wrong (wrong, obviously), they're pretty damn strict about using any related technology.
But that's non-canonical, and several of the comics have been contradicted by the sequel. In particular, the tribble comic's claim that Scotty was continuing to research transwarp beaming shortly before STID is hard to reconcile with STID's revelation that Starfleet/Section 31 classified and/or confiscated the technology -- something they would've had to do early enough to give them time to develop the portable transwarp-beaming rig Harrison used.