Advantage of being deuterostome
I would speculate that there is no direct advantage imparted by the deuterostomic development process over that of protostomes with regards to the evolution of more complex species. Instead, I would guess that sometime after that split, a series of novel alleles arose in various genes which were eventually fixed and laid the groundwork for future increased complexity. Given the trait in question (increased organismal complexity), I would be certain that these alleles were in mostly unrelated genes and were selected for some other function (in other words, there is no gene encoding a protein that imparts organismal complexity; that trait is, by definition, a complex one!). Perhaps the genes involved in the deuterostome/protostome split were part of this groundwork, given their activity at the base of development, however it would be very difficult to know at this point.



I don't know why I didn't think about this before, but the more likely explanation for the difference in complexity between deuterostomes and protostomes is actually unrelated to that difference. Most of the complex deuterostomes that you're probably thinking of are vertebrates. Vertebrates went through a couple of rounds of genome doublings (tetraploidizations) during their early divergence. The extra copies of all the genes provided fertile ground for evolutionary novelties that seems to have led to some wonderfully complex creatures. It's still possible that the tissue groundwork laid out by the deuterostome developmental process somehow contributes to that complexity, but it would be rather indirect, I think.