This point is so predictable I think it must be a favorite in Bible school classes. Actually an account of it on a temple wall would not be surprizing. Rameses II lost the Battle of Kadesh by his mismanagement but he devoted a huge wall at Karnak to it telling how he crushed his enemies beneath his heal. Spinning war reports didn't start with Bush.
Even so, there are those 500,000 papyri, mostly from the New Kingdom, covering every aspect of Egyptian life at the time. And there are the archives of neighboring countries, containing thousands of tablets, all silent on the subject.
There are other problems. From internal evidence it is clear that the story was composed in the 5th or 6th century, some 700 to 800 years after the time in which the story is placed. The author(s) knew nothing of the Egypt of the 19th Dynasty. The story does have the Hebrews working in the "store-city of Rameses" and that is sometimes said to be a reference to Piramesse. But Piramesse was Ramesses' residence and Egyptians did not have store cities, which were a Canaanite usage. Many of the subplots (Moses in the basket, the various miracles of Moses, even the parting of the sea) were common folk tales of the region, much older than Exodus.
I think it even turns up in the Gilgamesh writings. In fact, I think the same story was told about Gilgamesh, or something close to it.
You'll have to admit that it's an interesting possibility.
I'm a complete novice on the historical record, but I don't have to look hard to find something other than silence on the subject.
E.g, a much-discussed papyrus at the Museum of Leiden describing events paralleling the biblical account of plagues in Egypt mentioned here for example, and an inscription on a stele in Egypt translated as “Desolated Israel, that has lost its seed”, referenced here (Tomb of Mineptah).
You can try to dismiss or minimize such references, but don't pretend that they're nonexistent.