Virtually no biblical events are confirmed by other historical records until hundreds of years after the Exodus. The first time Israel was even mentioned in Egyptian records is in the Merneptah Stele (c. 1200 bce), which claims that "Israel has been laid waste."
The princes are prostrate, saying, "Peace!" Not one is raising his
head among the Nine Bows. Now that Tehenu (Libya) has come to ruin,
Hatti is pacified; The Canaan has been plundered into every sort of
woe: Ashkelon has been overcome; Gezer has been captured; Yano'am is
made non-existent. Israel is laid waste and his seed is not; Hurru is
become a widow because of Egypt.
Thus we have confirmation that a tribe or nation known as Israel existed in the time of Merneptah. But there is no mention of the Exodus or any details about who this group called "Israel" really was. There are tantalizing possibilities in the archaeological record regarding the presence of proto-Israelites or Hebrews in Egypt and the Sinai during the period in question. They include the Hyksos mentioned in the OP and also the Apiru spoken of in the Amarna letters. But these are speculative. For an in-depth discussion of this issue see Who Were the Ancient Israelites and Where did they Come From by William Dever.
But as the OP says, there is no mention of the crossing of the Red Sea in Egyptian records. Nor is there anything in the archaeological record to indicate the 40-year presence in the Sinai, of 600,000 Israelites of fighting age, plus another million or more women, children and elderly men.
Dever suggests that the reason for this is simply that the Exodus story combines a much smaller historical reality with a very large Mythic Truth. There were indeed proto-Israelite slaves in Egypt who escaped over a body of water [the Hebrew yam suph is "reed sea" not "Red Sea"), probably in several groups over a period of decades. Their escape was understandably not mentioned in the Egyptian records. These proto-Israelites later intermingled with groups such as the Apiru, Kenites, Midianites or other tribes on the outskirts of Canaan and eventually formed the Israelite tribal federation we read of in the Bible. The stories of their escape were embellished over hundreds of years of retelling around countless campfires.
This of course, is only a theory, but it does explain why there is no historical record of the Exodus outside of the Bible.