What you need is an operational method for interplanetary travel that makes sense and takes years which is different than an interstellar attempt. Using transfer orbits to be more fuel efficient and to increase useful payload would be one idea.
But any propellant-based, Newtonian propulsion system is fundamentally unapplicable for interstellar travel - and doubly so if the intent of the designers was to provide mere interplanetary travel. If they're pennywise with fuel and doing Hohmann orbits, they sure won't build any extra performance margin that would allow for an interstellar launch with minimal modifications.
You're taking about the small town that was adjacent to the missile silo that Cochrane was using? A dozen well constructed buildings, people warmly dressed, everyone looked fed, functioning businesses.
On the other hand, Cochrane
was off the grid, not being supervised by anybody openly associated with that USAF thing the launch hardware came from. And our heroes agreed on the particulars of the 2060s (and, in "Farpoint", also the 2070s at the very least) being post-apocalyptic.
Which is no reason to assume technology would be down. If the countries devastated by global war for real in the 20th century had one thing in abundance, that was technology. Not enough to win the war, but
that in turn was only because the other side had even more of it. Immediately after WWI, the world choked on automobiles and had a surge of aircraft to kickstart related enterprises. Immediately after WWII, it choked on aircraft and had a surge of spacecraft to kickstart related enterprises. It would only be fitting for WWIII to directly boost interstellar travel while making interplanetary travel a triviality.
Courts? We did see one such court.
Which was supposed to be representative of something. But what? The judge, the troopers and the spectators are decidedly Western in appearance, the few court lackeys Asiatic; should we think that, say, Britain had courts of this sort (to match the external trappings), or that the West took over Asia (to go with the surprisingly subdued role of Asia in the later fictional centuries)? Were courts like this prevalent in places that retained their power and technologies, or in places that had been weakened by the war?
Might be a place run by judges like that would get warp engine projects going, and then
keep them going, even if it cost a few hundred thousand lives. MIght be such projects would go off the grid after a fashion, channeling the profits and benefits to the select few.
Or, the Botany Bay was much less than one light year from Earth, only having made it as far as the outer solar system. This is where the Enterprise found it.It fell into what used to be called a "black hole."
The massive problem with that is the lack of effort by our heroes to take Khan to Earth. Their authority of choice is Starbase 12, and they fail to reach that within plot time. (Additionally, Khan's CQ would surely be overheard by others, too - but arguably it's a recent thing, triggered by Kirk's chance approach. Yet that in turn suggests no flights to the region in the recent past, which is rather unthinkable for the general vicinity of the most important UFP hubworld.)
Khan reaching a distance of 100+ ly (and, consequently, a relativistic speed to account for the odd references to passage of time) would be the more palatable model to go with the SB 12 thing. An "eternal" engine, free of fuel or propellant consumption issues, would be the way to attain that - and we can give Khan that type of engine, as long as we keep the performance down inside Sol.
As seen above, going for a single fixed acceleration is problematic. Could we boost acceleration outside Sol somehow? Well, we already know that warp drive consistently becomes incredibly sluggish close to Sol (and perhaps other stars, too) - we see with our own eyes warp 10 equating sublight in ST4:TVH, say. Perhaps Khan's drive, too, was stuck in tar when operating close to Sol, but would start performing much better when passing the heliopause or duonetopause or whatnot?
This unexpected change in performance would also nicely match the official reason for the loss of the
Charybdis - a thrust anomaly! (Alas, that particular Okudagram never made it on screen AFAIK, and can only be seen in the backstage books...)
Later generations of starfarers would be aware of DY-100 ships having reasonable interstellar performance even when their insystem performance remained abysmal. This would allow for automated ore transport use, and keep our heroes' expressions of surprise subdued in the "Space Seed" teaser, while still precluding conscious attempts of using DY-100 for crewed flights to the stars back when the ships were still a thing.
Heck, Khan may have been propelled to the stars wholly accidentally, too! Colonizing an off-Sol planet was never part of his plans - he's as surprised by the idea as anybody when Kirk forces it down his throat. Going out to the Oort Cloud and chilling it out there in cryosleep while the trail goes cold would be a cool maneuver for our icy-hearted villain, if you catch my drift... The engine suddenly going to overdrive would ruin the perfectly good plan, is all.
Timo Saloniemi
Excuse me while I go barf. 
