It sounds more than a bit unlikely that Starfleet would lack an edge over the Klingons if it has enough ships to spare that one of them would "stumble" onto Khan...
That doesn't mean they wouldn't still seek one, especially with the preemptive militaristic mindset Admiral Marcus represents.
Which sort of begs a fairly interesting question: assuming his present disposition isn't also a consequence of the loss of the Kelvin -- and I don't see how it would be -- then what the hell was Marcus like in the Prime Universe?
My suspicion is that the likes of Admiral Cartwright and Colonel West were probably among his students after he retired; I'd even go so far as to say that the Klingon-Federation War that was eventually spoiled by Organian interference might have been Marcus' doing in the first place, that the FEDERATION actually started that war and it was his involvement in initiating the hostilities that lead Carol Marcus to leave Starfleet and look for civilian work.
And how would that ship know that Khan was worth anything?
The same way Kirk knew: when the commanding officer finally figured out who Khan really was. The only difference in this case is the CO of that ship was a (more paranoid than usual) Section 31 affiliate and accordingly didn't give Khan unrestricted access to the ship's library computer.