For example, which Pharaoh would have been most likely, I have been given the suggestion that it may have been Amenhotep III, given his large building project and the death of his first born son. What is the most likely period for this to occur?
Well, Bible itself gives a good hint:
Therefore they did set over them taskmasters to afflict them with their burdens. And they built for Pharaoh treasure cities, Pithom and Raamses. (Exodus 1:11)
Ramesses II of the Nineteenth Dynasty build himself a new capital in Nile delta, named Pi-Ramesses, to have better control of Syrian affairs and such large scale constructions fell as heavy burden of local population .