It should be noted, with the context of DSC now added in: Whatever it was Garth of Izar did at Axanar - and against whom (clearly now it wasn't the Klingons) - happened before the current series. If Kirk was just starting out as a cadet when the battle and then peace mission occurred, it would have been five years ago now in DSC's 'present' time.
		
		
	 
It should have already taken place with respect to DSC, absolutely. And most probably shouldn't be against the Klingons, no.
A few small quibbles, if we 
really wanted to drill down 
@Timo-style and wiggle as vigorously as humanly possible to pry open and preserve the most breathing room that can be had for ourselves...
I'm not 
entirely sure that in context "new-fledged cadet" 
couldn't mean recently-
graduated cadet, i.e. it actually 
could have been green Ensign Kirk just 
out of the Academy who went to Axanar
—perhaps aboard the 
Republic alongside Ben Finney and under the command of the same Captain Garrovick with whom Kirk would later transfer to the 
Farragut for his first true "deep space assignment"—after having studied Garth's preceding victory while still there. If so, that 
could make it as little as two years or so before DSC. If 
not, then another possibility could even be that the peace mission in question actually 
preceded the battle there, and Kirk had 
already been there on this little class field trip 
before he studied Garth's subsequent victory, with the latter becoming a hot topic as a current event after he'd gotten back.
Anderson in "The Vulcan Hello" mentioned some "fleeting run-ins" with Klingons, so it's 
barely possible that Axanar could have been one of those, so long as (almost) no one who lived to tell the tale actually 
saw a Klingon there, as opposed to just their ship(s). Or, since no specific opponent was mentioned, maybe Garth didn't even 
know who he was fighting?
I do not 
favor either of the above contortions as they seem ungainly and unnecessary, but they are perhaps 
just manageable should need arise. I have a post in the works for another thread with a more straightforward outline of Kirk's early service record as referred to in TOS, with all the complete quotes and references, as I've been re-watching and making a new attempt at it. Stay tuned...
In any case, DSC 
isn't heading for Axanar, barring non-linear storytelling, time travel shenanigans, or major retcons.
But Lorca 
could turn out to be Garth...


 
(Don't write me letters.

)