If you stood and fought the innocent could have ran away or hid. There was a chance you fighting the dragon gave the innocent life.
There was a chance that if you danced a jig then a meteor would strike the dragon dead.
So, "chances" dont really help the discussion much given what we know.
"He was carrying an injured NPC to safety."
Unless we assume stoopid, the injured guy was worse at getting away on his own. Otherwise, why carry him? Was the oath made to Lyft?
"Unfortunately, an adult dragon cornered him." Adult dragon solo vs 7th level PC is not winnable unless extremely odd situation.
"Give me that man, and you can live. I hunger" it said."
Note no offer to take the paladin and let the injured live.
"He figured he should live to fight another day (and continue on the world-saving adventure the party are part way through)."
So, on top of this seemingly unwinnable situation, there is a literal world-at-stake quest involved., already engaged.
So, "cornered" and "carrying" both leave me to not really see any sort of dign that the fight letting the guy escape thing you imaginatively added was actually part of the scene. It certainly did not make the OP notice it enough to mention it.
Again, to me as described, the "service to good" option here is the one taken - accept failure in protecting this injured noc, survive, then save the world and return to dlay dragon and rezz the fallen in ehicheverborder makes sense.
There is a difference between trying and failing and breaking your oath.
The "carrying an injured NPC to safety" failed when the adult dragon caught them. Thecredt was just the consequences of that failure.
Unless of course "throw your life away for pride" is part of his oath.