So here's the answer, for anyone else wondering..
Eventually the amount of energy possessed by a solution with excess negatively charged ions will overcome the high energy input required for the 2 d-shell electrons of Pb to create covalent bonds alongside the 2 outer p-shell electrons. So instead of the solid PbI2, which only takes care of 2 I- ions, the covalent, aqueous molecule PbI42- forms and lessens the amount of excess energy in the solution.