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I think you're referring to The Transcendental Argument for the Existence of God (TAG). It was first formulated by Immanuel Kant, but has been refined, disputed and defended over the years.
One formulation of the argument is,
Prove A: The Christian God exists.
Step 1 ~A: (Assume the opposite of what we are trying to prove): The Christian God does not exist.
Step 2 (~A--> B): If God does not exist, then there is no intelligible experience since God is the precondition of intelligibility
Step 3 (~B): There is intelligible experience (Contradiction!)
Step 4 (~ ~A): It is not the case that God does not exist (Modus Tollens on 2 and 3)
Step 5 (A): --> God does exist (Law of negation.)Source: Reformed Apologist (blog)
The finer details of whether this argument actually is valid, of course, need a more thorough explanation. The argument isn't fundamentally based on inductive nor deductive, but transcendental reasoning. Kant describes transcendental as follows:
I entitle transcendental all knowledge which is occupied not so much with objects as with the mode of our knowledge of objects insofar as this mode of knowledge is to be possible a priori.
Thus to properly examine and understand the argument, I suspect a more in-depth studying is required. I'm not a philosopher and don't claim to understand TAG, so I'll just point you elsewhere:
Butler, Michael R. The Transcendental Argument for God's Existence.
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Indeed, for logic and any other knowledge to exist, there should be directly experienced thing that does not need proving. Yet it is not evident that this thing should necessary have properties, which are attributed by Christians to God.
As metaphysics tells us via Anthropic principle, it is existence of self that is needed for logic and any experimental facts to exist. In other words, for the universe to exist there should be an observer, for logic to exist, the observer should be conscious.
It is possible to compare the properties of the self and Christian God.
Christian God creates the Universe consciously.
The self while necessary for Universe to exist, has limited abilities to affect it consciously.
Christian God is almighty
The self is not almighty definitely. It has limited abilities to affect the things in the universe (freedom of will) though. However, such ability guarantees that the Universe is neither deterministic nor random (free will is incompatible with both determinism and randomness, see Peter van Inwagen).
Christian god does not obey physical laws
The self also does not obey the usual physical laws. Even more, there can be no universally valid physical theory, that is theory that correctly describes a system that contains the self (see Thomas Breuer). Yet this disobedience has own limits: on practice it is usually manifested by the existence of principally non-measurable parameters (subjective decoherence, see Breuer).
Christian God is immortal.
It is unknown whether the self is immortal. Either case, death of the self means disappearance of the universe. There are suggested quite plausible mechanisms through which the self may be immortal.
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that in order for logic to be valid, one could not have circular arguments...
Bad Logic does not prove that logic does not exist. At the very least all it proves is that bad logic exist.
but since you can't prove your premises, all arguments had to be unfounded or circular unless you introduced God as some sort of self-proving entity.
This is a very broad generalization. Why can't the theist prove their arguments. Kalaam Cosmological argument seems like good proof to me. It is a in depth discussion, but to think Christian notion of God is circular would need quiet a bit of explaining. (to say the least)
(e.g. why it is okay for God to effectively be a circular argument but not for anything else to be)
It would not be good for it to be circular, but where has the Christian God committed circular logic? The skeptic needs some argument to support that view because I'm not aware of any argument for God's existence that is circular.
Does this argument have a name, and is it (or at least a summary) written somewhere?
Not that I'm aware of. It is the first time I have heard of someone thinking the notion of a God commits circular logic.
I'm not quiet sure how to answer this objection because the OP makes some assertions but does not back it up with something substantial to support the claim
I suspect that some strawmanning of arguments are to blame for this