It would take a lot of work, but they would conclude it sooner or later. When Saul arrived at her room to talk about Yevgeny she could assume that he had got knowledge of the content of the recording. If she had been more open about her talk with Yevgeny right away, the outcome would have been the same, but she would have had more time to talk about why she wanted to talk to him.
I understand what you’re saying, but I think you’re looking at this from an extreme utilitarian viewpoint (“what would have been the most effective way to make her argument”) and not from a more human, emotional viewpoint that takes into account how humans react when they’re caught in a lie.
I think we’ve all been in a situation where we’re caught in a lie, we know we’re caught in a lie, but we don’t come out and admit the truth. We just keep digging.
You’re expecting Carrie to act very logically and un-emotionally in this scenario when by its nature it’s anything but. The point is that she is caught up in her emotions about this man and what he might or might not mean to her. And she can’t explain it yet either because, in Carrie’s own words, she’s still “putting the pieces together.” To admit that not only to herself but to the person she probably trusts most at the moment is too fucking scary.