To be clear, since Pharoah’s never mentioned their defeats, there is no reason why a large Exodus couldn’t have taken place. Eric H. Cline (whose book 1177 BC is an excellent read for anyone interested in the ANE and how it is still relevant today) agrees:
We do not have a single shred of evidence to date. There is nothing [available] archaeologically to attest to anything from the biblical story. No plagues, no parting of the Red Sea, no manna from heaven, no wandering for 40 years. However, I should add that there is also no archaeological evidence that proves it did not take place. So at this point in time, the archaeological record can neither be used to confirm nor deny the existence of the Exodus.
Friedman, Richard Elliott. The Exodus (p. 20). HarperCollins. Kindle Edition.
Still, I think there is good textual evidence for an Exodus of Levites only.