<Khan> Time is a problem you don't have, Admiral... </Khan>
There's no indication how much time passes between Kirk's trial and the day he gets the E-A. Could be years for all we know. Certainly there's no indication it would be mere days.
In addition, the Saratoga didn't appear to have damage that extreme from the probe's effect.
was completely destroyed and everybody aboard killed for all we know. Just because her exterior was pristine doesn't mean a thing.