Why was the last king of Saxony styled as Friedrich Augustus III, instead of Friedrich III or Friedrich III Augustus?

Probably because his great-uncle had already been styled Friederich-August II, and his great-great-uncle, the elector Friedrich Augustus III, choose to be styled Friedrich Augustus I when he was upgraded to king.

I know that it's been customary for Catholic and Protestant royals to be christened with multiple names (unlike Orthodox royals). But if a sovereign is not going to style himself by his entire name ("Friedrich Augustus Georg Wilhelm ..."), why use two, instead of just one? Doesn't this make things messy for his successors?

Suppose the kingdom of Saxony had survived, and the eventual successor been Prince Friedrich Christian? Would he have been known as King Friedrich Christian I? Wouldn't it make more sense for him to be King Friedrich IV, and his predecessor King Friedrich III?

If the kingdom of Saxony had survived, probably F-A III's heir, Pr.Georg, would not have abdicated his rights and would have become king Georg II.

I personally like the Swedish way: the present sovereign is King Carl XVI Gustaf, his grandfather was King Gustaf VI Adolf, etc.

Agreed.
If the law hadn't been changed, Carl XVI Gustav would have been succeded by Carl XVII Filip.

For this reason, I think it would have made sense for the longest reigning emperor in Habsburg history to style himself as Franz II Joseph, as opposed to Franz Joseph or Franz Joseph I. As I understand, his grandfather was the last Holy Roman emperor (known as Franz II) and first emperor of Austria (known as Franz I).