Can I chose to bring a shockland untapped into play when I tutor it with an effect that brings a land tapped into play
No.

If you read Godless Shrine's ability carefully, you'll notice that it doesn't say the card enters the battlefield untapped if you do pay; it says the card enters the battlefield tapped if you don't pay.

Godless Shrine only affects how the card comes into the battlefield when you don't pay; it doesn't affect how the card comes into the battlefield when you do pay.

Flagstones of Trokair causes Godless Shrine to come into play tapped, and Godless Shrine doesn't do anything about that (even if you do pay).