It's a Natura2000 zone, they did build illegally without a permit in the first place, and argue that a retroactive permit could be given because of the small footprint.

Two tribunals at two different levels (additional authorities separate from the mayor) judged that they were wrong.

I would argue that you did not explain the actual situation honestly.

> Well, isn't "following construction rules" basically means "use concrete and established plans"?

It does mean that you need to submit plans beforehand. Concrete is absolutely not a hard requirement in France.

But building in a natural protected zone without a permit is asking for trouble. Maybe a small wooden shed would have been ok.