Shadow Gunner Level select
At the main menu, hold Select and quickly press Triangle(3). Then, press Square, X, Square, Triangle, X, Triangle, Circle, Square. Begin a new game, then hold Select and press Left or Right at the mission introduction screen to choose a new starting level.
Invincibility
At the main menu, hold Select and quickly press Triangle(3). Then, press Triangle, Circle(2), Triangle, Circle, Square, X(2).
Super gun
At the main menu, hold Select and quickly press Triangle(3). Then, press X, Circle, Triangle, X(2), Circle, X, Square.
Invisibility
At the main menu, hold Select and quickly press Triangle(3). Then, press Square, Triangle, X, Square, Triangle, Circle, X, Triangle.
Overhead view
At the main menu, hold Select and quickly press Triangle(3). Then, press Triangle(2), X(2), Square, Circle, Square, X.
The image featured at the top of this post is ©The Legend of Heroes: Trails of Cold Steel III key art.