How do I make an actor's character change when they are dead?

What is the most efficient way of doing this? Editing anything in the troop events is not an option.

A small script can do it without touching any events. Open the Script Editor (F11), insert a new slot under ▼ Materials (above ▼ Main) and paste this:

class Game_Actor < Game_Battler
  alias dead_graphic_character_name character_name
  alias dead_graphic_character_index character_index

  def dead_graphic
    return nil unless dead?
    return nil unless actor.note =~ /<dead graphic:\s*(.+?)\s*,\s*(\d+)\s*>/i
    [$1, $2.to_i]
  end

  def character_name
    graphic = dead_graphic
    graphic ? graphic[0] : dead_graphic_character_name
  end

  def character_index
    graphic = dead_graphic
    graphic ? graphic[1] : dead_graphic_character_index
  end
end

class Game_Player < Game_Character
  alias dead_graphic_update update
  def update
    dead_graphic_update
    dead_list = $game_party.battle_members.map {|member| member.dead? }
    refresh if dead_list != @dead_graphic_list
    @dead_graphic_list = dead_list
  end
end

Then add a tag like this to the Note box of each actor in the database, using the file name from Graphics/Characters and the character’s index on that sheet:

<dead graphic: Coffin, 0>

On a regular sheet the index runs 0 to 3 across the top row and 4 to 7 across the bottom. A $ sheet with one character is just 0. While an actor is knocked out, the player or follower shows that graphic, and it switches back on its own when they’re revived. Actors without the tag keep their normal graphic.

Oh my god you went the full nine yards. Thank you so much.