I'm not a presuppositionalist myself, but the respective works of Gordon Clark and Cornelius Van Til (in addition to Bahnsen, Frame, et al) are all thought-provoking. One of the influences of St. Augustine on Clark is the claim that the use of logic presupposes God's existence by the very nature of logic itself. Augustine uses the example of 2+2=4, whereas Clark claims that the laws of logic correspond to the divine attributes.
Let's take the example of the transitive axiom: if A=B and B=C, then A=C. This axiom, expressed as a proposition, is objective and unfalsifiable. It holds at all times, which means it is eternal. It also holds in all places, implying its omnipresence. The truth of logic cannot be changed into its contradictory, or negation, so it follows that it is changeless. Finally, contradictory systems of logic cannot both be true; this means that logic is one. In sum, we know that logic possesses the attributes of eternality, omnipresence, immutability, and unity. This already sounds rather God-like. Left without further analysis, however, we're left with a merely impersonal God, something like a vague principle of intelligibility.
If we combine this proof with conceptualism, on the other hand, then we can show how it makes sense to postulate the existence of a necessary mind. This would be a sort of "closing-of-the-gap" between logic and God. After all, the Logos referred to in John 1:1 is logic itself (Clark even translates John 1:1 as: "In the beginning was the logic, and the logic was with God, and the logic was God"), but it is also personal. We're often in the habit of treating logic as if it were this impersonal abstract principle, but in conjunction with conceptualism, we can show how the use of logic can be part of a rationally compelling proof of God's existence. We can summarize the argument as a reductio ad absurdum:
Prove A: God exists.
Assume ~A: God does not exist.
~A --> B: If God does not exist, then logic is not objective.
~B: Logic is objective.
Hence, ~~A: by modus tollens.
Therefore, A: God exists.
Of course, the proponent of this argument will have to argue in favor of both realism and conceptualism.