Who was/were the king/s in Egypt for the duration of the Jews' exile there?

Question

In particular, who was the Pharaoh on the throne during the actual exodus?

I would be interested if anyone has any knowledge of how Egyptian historians correlate information they have from artifacts, drawings, writings from that period to what we know from Torah sources.

Here is a list of Egyptian Kings I found. I have highlighted two kings - between which the exodus happened. It is possible that the actual Pharaoh who was sent the Macos/let the Yidden go etc, is not on this list - the Egyptians would not record in their lists of Kings any that did not conform...and 'our' Pharaoh must have been an embarrassment to them. something to note is the very short lifespan of the kings (so during the 210 years the Yidden were in Mitzroyim there could possibly have been many Pharos. Note the King Haremham (also highlighted) who had a fairly long reign and was the last of the 18th Dynasty (could he have been 'our' Pharoh)?

Year 2448 was yezius mitzrayim 1312 B.C. So yidden were in mitzroyim from 1522 B.C (210 years before).

Dynasty 17 Tao I ca. 1560 B.C. Tao II ca. 1560 B.C. Kamose ca. 1552–1550 B.C.

Dynasty 18 Ahmose ca. 1550–1525 B.C. Amenhotep I ca. 1525–1504 B.C. Thutmose I ca. 1504–1492 B.C. Thutmose II ca. 1492–1479 B.C. Thutmose III ca. 1479–1425 B.C. Hatshepsut (as regent) ca. 1479–1473 B.C. Hatshepsut ca. 1473–1458 B.C. Amenhotep II ca. 1427–1400 B.C. Thutmose IV ca. 1400–1390 B.C. Amenhotep III ca. 1390–1352 B.C. Amenhotep IV ca. 1353–1349 B.C. Akhenaten ca. 1349–1336 B.C. Neferneferuaton ca. 1338–1336 B.C. Smenkhkare ca. 1336 B.C. Tutankhamun ca. 1336–1327 B.C. Aya ca. 1327–1323 B.C. Haremhab ca. 1323–1295 B.C.

Dynasty 19 ca Ramesses I ca. 1295–1294 B.C. Seti I ca. 1294–1279 B.C. Ramesses II ca. 1279–1213 B.C. Merneptah ca. 1213–1203 B.C. Amenmesse ca. 1203–1200 B.C. Seti II ca. 1200–1194 B.C. Siptah ca. 1194–1188 B.C. Tawosret ca. 1188–1186 B.C.

Answer 1

R' Hertz writes in his commentary on the Chumash that, while the identities of the various malchei Mitzrayim cannot be definitively ascertained, there is a sufficient historical record to pinpoint the events of our time in Egypt.

He writes that when Yosef, Ya'akov, and the rest of B'nei Yisrael came down to Egypt, the rulers were the Hyksos kings. He continues that when the Torah says ויקם מלך חדש על מצרים וכי׳, it refers to the New Kingdom, which was ruled by ethnic Egyptians. He continues that we, as high-ranking members of the Hyksos nobility, were enslaved and points out historical documents from individual Pharaohs about 'Apiru (עברי in Hebrew) being used to build monuments.

Answer 2

Rabbi Dr. Elihu A. Schatz goes through this in depth with many fascinating bits and pieces of remaining Egyptian history in his book " Proof Of The Accuracy Of The Bible". I cannot go through all the info that he details since its a build up from the beginning of the book how the dates fit in with each era,but will summarize the main idea.

He writes on pg.54 that Thutmose III was the Pharaoh who did not let the Yisraelim leave Egypt. He notes that besides a Pharaoh there was also a king of Egypt and that was Hatshepsut. After Hatshepsut died Thutmose III took the role of both. To learn why he came to these conclusions one would have to read the book which I found very interesting to say the least.