The most idiomatic would be:

J’ai pris une douche ce matin.

Your first suggestion is correct and idiomatic too:

Je me suis douché ce matin.

but might carry a small nuance. J’ai pris une douche ce matin just states a fact while je me suis douché ce matin might suggests doing it in that morning was unexpected (e.g.: you do it usually on the evening) or questioned. I might be overthinking it though…