Yes, that works.

From the Comprehensive Rules:

122.1d A stun counter on a permanent creates a replacement effect that stops the permanent from untapping. That effect is “If a permanent with a stun counter on it would become untapped, instead remove a stun counter from it.”

What's notable here is what it doesn't say. Stun counters prevent a creature from untapping. They don't prevent it from doing anything else that it would normally have to tap to do, if you can find a way to do that thing that doesn't require tapping it. Doors of Durin is just such a way.

A vaguely related rules interaction: I had a game recently where I was able to attack [[The Eternal Wanderer]] with two creatures (despite its static ability that disallows that) because one of them was a token from [[Adeline]].

Doors of Durin says:

Whenever you attack, scry 2, then you may reveal the top card of your library. If it’s a creature card, put it onto the battlefield tapped and attacking.

This means the creature enters as an attacking creature, rather than being declared as an attacker. Watcher in the Water does enter with stun counters on it, but those don't affect the fact that it already entered as an attacking creature.

Stun counters only prevent a permanent from untapping. That's all they do.

So yes, it does the flavorful thing.

• Doors of Durin - (G) (SF) (txt)
• The Watcher in the Water - (G) (SF) (txt)

[[card name]] to call

That's a cool flavour win!