God as a non-empirical being

[Mudhammam]

It seems to me, that a theist might respond that God is able to act within the Universe and that the effects of God's actions can be experienced or observed, but that the God itself remains outside the realm of experience or observation. But again, I can't think of any rational justification for making the claim. The sole reason seems to be to absolve the claimant from having any burden of proof.

Exactly. In that case, the person not only always fails to provide evidence for their claims, but more importantly, they're arguing in a logically invalid manner. There's no rational justification that "God exists" on the basis that some event occured for which we simply have no explanation, and otherwise known causes means we can point to a preceding event through which the known laws of physics or chemistry or biology are precluded as necessary or sufficient conditions for the consequence.