What is the accuracy of scf=qc when convergence criteria is lowered?
Something that I have discovered is that when Gaussian (or other good ab initio software) fails to converge, there is something sketchy about what one is trying to calculate. Sometimes DFT calculations just aren't appropriate, especially in cases that require multiconfigurational SCF for even a zero-th order description.

In this case, given the sheet-like character, I suspect the problem is that the molecular orbitals are so close together, and so similar, that there are serious linear dependency problems that lead to all sorts of havoc in the linear algebra routines and convergence accelerators.

If you are trying to simulate an infinite sheet, then I suggest that you try plane wave codes like those found in solid state chemistry (Gaussian's PBC code, NWChem, abinit, etc.)  These sorts of calculations have gotten easier in recent years, but sometimes they require serious horsepower.

There really does seem to be a problem with SCF convergence, and this could be because you are approaching some kind of band structure limit. I have always used QC as an absolute last resort when I must do something wacky.

(One other thing, you have "capped" your peripheral benzenoid rings with hydrogens?)