It makes sense that you would be able to tell in someone was speaking to you through a universal translator.^ The kind of civilian pilots the Enterprise crew were impersonating, would not have a translator. So if the Klingons knew they had one, the game is up.
I'm pretty sure that we briefly heard a Stephen Hawking voice coming out of the hand unit Kirk was holding to his ear during the court case before the reality softened to allow us to listen in English.
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