@demoskinos: Assuming your progress on that quest doesn't carry over into new game plus, (nothing majorly spoilery here, but gonna mark it just in case): the secret dungeon seems to house only Last Ones, so I'm guessing that's the way you're supposed to actually beat all of them in one playthrough
Your progress does carry over into NG+ since it's looking at your bestiary completion (which you keep) rather than the monsters you've eradicated (which resets). That's why the guy who gives you the quest says weird things like "maybe this would be possible for someone who lived more than once".
You're right about the secret dungeon being an easy way to wipe out most species, but unfortunately it's missing all the ones you face in the final dungeon (Chimera, Set, etc.) and you can't back out once you're in there, so even if you did it on a single playthrough you'd need to start a NG+ just to turn it in.