My understanding is that the function of TAs is not to derive or deduce God at all. Rather, simply to point out that things like deduction itself presupposes Him. There is no neutral ground from which deduction can begin its journey. To begin a deductive trail of thought is to have already granted God's existence, as well as several other things. TAs observe that these presuppositions are tied up in the nature of deduction.
Circularity does indeed seem unavoidable.
The charge of circularity only works if we allow ourselves to fall into the trap of thinking that all beliefs require logical proof or argument. The very concept of a presupposition is the recognition that not all beliefs require such argument. What we might argue, after the fact, is that having presupposed LC, we can bear the fruit of having done so, whereas others can't. TAs seem to do less debating over the raw materials of worldviews, and instead simply offers a challenge to the opponent to try to build a stable house out of their chosen materials. Or it asks for an accounting of how the house can be stable given the materials allegedly used to build it.