For a discussion of basically the same topic before, I recommend checking out this discussion thread between myself and /u/kabrutos .

In sum, the arguments tend to conflate O-explaining and E-explaining, due to conflating the order of being with the order of knowing. To O-explain is to provide an ontological grounding for (i.e. the efficient cause of, the material cause of, the supervenience upon, etc), whereas to E-explain is to provide an epistemic justification (i.e. a reason to believe and so forth).

With regards to E-explanation, neither the atheist nor the theist (let alone specifically the Christian theist) enjoys an advantage - both sides will take a certain class of foundational beliefs to be more or less non-negotiable, insofar as the rational project is concerned and neither side can E-explain those beliefs, as to E-explain will itself require use of them. (This is probably clearest with something like the use of deductive or abductive reasoning, although I'm personally not convinced that there aren't decent probabilistic arguments in defense of induction - see Vitanyi and Li's paper here for an example, at least insofar as 'straight' solutions to inductive skepticism is concerned) As /u/kabrutos notes, "Just take whatever it is that God's existence is alleged to create--'epistemic foundations'--and say that you believe in those things."

On the flipside, with regards to O-explanations, the arguments will turn on various classical evidential arguments. For instance, the appeal to logic here will simply a variant of the conceptualist argument, as put forth by luminaries such as Augustine or Leibniz. But, this is no attempt to E-explain logic, In particular, if the presuppositionalist acknowledges that neither they nor the atheist can E-explain logic, but insists on the view that only the theist can O-explain logic, then there is no longer a distinctively presuppositional argument, but a strain of a classical/evidential argument, such as Leibniz's argument from eternal truths. And of course, the atheist can challenge either the claim that the theist can invoke God to ontologically ground the phenomenon in question (necessary truths of logic or morality, for instance) or the claim that there is no plausible nontheistic alternative (such as varieties of platonism or nominalism). Indeed, arguably it will take nonetheless - in my view at any rate - that a sort of total divine simplicity to elude the Euthyphro-esque concerns that plague the first route (that God ontologically grounds such necessary truths), a move that's going to be incoherent at worst and highly implausible to many atheists and even many theists at best.

As such, I don't think there are high hopes to be had for presuppositionalist apologia.