No, you can't do that. Short version: the Ancient One devours you as soon as you're reduced to 0, so by the time the Injury/Madness tries to change how you recover after your're reduced to 0, you're already gone.
In combat with the Ancient One, you're devoured if you're reduced to zero Stamina/Sanity:
Any investigator reduced to 0 Sanity or Stamina by the Ancient One’s attack is devoured. If an investigator is devoured during the final battle with the Ancient One, that player is eliminated from the game (i.e., the player does not draw a new character).
You're devoured, not just knocked unconscious or driven insane. You don't get to choose which way you want to get knocked unconscious (item loss vs Injury), because the Ancient One ate you.
The rules for Injuries (Madnesses are similar):
When an investigator is reduced to 0 stamina, he may choose to gain an Injury card. If he does this, he does not lose any items or Clue tokens for being knocked unconscious, and his Stamina is restored to its maximum value instead of to 1. However, the investigator must still move to either St. Mary's Hospital or Lost in Time and Space, as appropriate.
You still get your Sanity/Stamina reduced to 0 - you just recover afterwards in a different way. So the Ancient One text still applies, and you're still devoured. You don't have the painful recovery (recover to 1, lose items, etc) afterwards that you can replace with the Injury/Madness. You can't take an Injury any more than you can say, "okay, he got me - I'll drop my useless item and keep on going!" Getting devoured is permanent.
Finally, this is just common sense. If adding Injury/Madness to the game somehow let you stay in the fight, then the Dunwich Horror expansion would make the final battle way, way easier. The expansion isn't meant to make radical changes like that.