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Would it be possible to export a character with different modules?

For example, if I picked a hat for my character to wear, could I have the exported sprite sheet be something like:

Row 1: Character's walk cycle
Row 2: Character hat's walk cycle

That way I could allow my players to choose if they want to wear a hat or not.

If this is possible, it could be expanded with different faces, facial hair, hats, etc.

You can just export a spritesheet of the character without a hat and with a hat, how you implement that in your game is up to you though.