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