JP Morgan believes that he was once an Egyptian Pharaoh. Whether he is just a senile old man searching for the answer to one of life's biggest questions, or actually the reincarnation of an ancient Egyptian Pharaoh, Morgan is convinced that he is been reincarnated, and will be again following his death, to lead the world forward. Morgan believes that only the very best and the brightest are the reincarnations of Egyptian Pharaohs, for example Henry Ford. However, during his disheartening experience inside the pyramid, his belief turns into a coping mechanism, as he realizes that he may not be a reincarnated pharaoh. However, by introducing the concept of reincarnation, Doctorow lets the reader
While he is in the pyramid, Morgan waited for a sign from Osiris so that he can connect with his Egyptian brethren. With his dream about being a simple commoner in ancient Egypt he seems to go into denial and claim that it was a false sign. I think it's worth noting that in Egyptian mythology, in order to enter the afterlife and be reborn your heart must be pure and free of sin. I'm not sure if Morgan would be considered free of sin given "acquisition" of art and the sarcophagus of Seti I (I'd assume it would be considered a sin to steal a dead pharaoh. I think this alone disqualifies him from rebirth.).
On the other hand, Doctorow alludes to Coalhouse Walker being a reincarnation of the literary character Michael Kohlhaas. I think it is fair to say that Coalhouse is a direct reference to Michael Kohlhaas (though the average reader would likely not deduce this), as they both face similar challenges and meet a similar fate. Considering that the Michael Kohlhaas story is an adaptation of a real life event involving Hans Kohlhaas, it seems as though Doctorow is suggesting a reincarnation of the Kohlhaas/Coalhouses to fight against their respective injustices. In this sense Doctorow is contradicting Morgan's view of reincarnation, that only the important people get reincarnated, instead suggesting that the "pure of heart" per se, who fight against injustices deserve reincarnation.
To again relate to the Egyptian mythology of rebirth, Coalhouse is seen as more deserving than Morgan. While Coalhouse does commit arson and kill several firemen, and I do not condone his actions, given that he was fighting for what is right, it could be seen that his heart is pure. I admit that this may be a stretch to the aforementioned logic regarding Morgan in relation to Egyptian mythology, but given Doctorow's love for subtle hints, I feel like it is a valid stretch.