I've been trying to unlock the Burst Shot Ex Skill for a few days now. On the surface my rank should be good enough. I'm on normal mode, I got an A Ranking, not sure what else to do.
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If my memory serves right, the screenshot you included only proves that your rank was A after you completed the mission.
To obtain an EX Skill from a boss you have to fight that boss while being at A/S rank. In that case the boss will start using their version of the EX Skill as a new attack and when you beat them they will give you the skill.
Therefore, you should check that you rank was high enough when you entered the stage and not just when you completed it.