Let's for arguments sake assume that the singularity is the first material substance to come into existence. This means that there was no material substance any moment before it, in fact there were no moments before it. It then follows that it does not have an accidental cause i.e. a cause that is prior in time to the singularity. So the question "what is the accidental cause of the singularity" is simply nonsensical.We know that the observed universe started with an expansion event (about 14 BYA), we do not know the origin of the singularity from which the observed universe expanded.
Does this imply that the singularity has no essential cause? No. Why not? Well, if its nature was its act of existing then it would not cease to exist. But it did cease to exist. The singularity is no more. This implies that it was a contingent material substance and therefore that it was a composite of its essence and its act of existing. This implies that something conjoined its essence and act of existing whenever it existed and that essential causal chain ends in something whose essence is its act of existing (just like for any other material substance).
The universe is the sum of all material substances. The beginning of the first material substance heralded the beginning of the universe (for arguments sake, the universe began with the beginning of the singularity). The existence of a universe depends on the existence of material substances. The first material substance (the singularity) was contingent, thus the beginning depended on the existence of a first contingent material substance, which ultimately depended on something whose essence is its act of existing.No, seems to me it would be necessary, and everything within it would be contingent. (That neatly does away with the necessity of your second tier necessary being)