The problem here is, that the bible is essentially our only (more or less) reliable source for Moses - and even that was written down centuries after the supposed events.

There is no single Egyptian source that mentions Moses or an event similiar to the Exodus. Interestingly enough "Moses" is an Egyptian name, a shortform of theophoric names like Ramesses, deriving from the verb msj ("to give birth", therefore: Ramesses, ra-msj-sw, "It is Ra, who was born him"). From the perspective of Ancient Egyptian sources something like the "Exodus" has never happened though. We know of the presence of so-called "Asiatics" in Egypt in the New Kingdom and we have some indications in certain sources, that there arose conflict with some of these "Asiatics" in the later New Kingdom. I have written about these sources and possible links between the biblical Moses and some semitic figures mentioned in Egyptian sources here. But keep in mind that this is effectively nothing more than educated speculation, there's no definite proof that these links were a reality. As I have written in the linked comment, it's most probable that the biblical stories reflect a reality that was already centuries away when they were finally written down. They are a distorted reflection, a memory of earlier days - like so many of our still high valued myths today (King Arthur, Homer etc.). None of these are entirely ficition but always reflect a past reality through the lense of the then contemporary reader.

The linked comment talks much about Egyptian sources and presents some educated guesses from the modern-day literature, but let me talk about more about "the Jews" in the time of (a supposed) Moses. The bible requires the existance of a fixed ethnos, called "Hebrews", that was more or less complete present in Egypt, was subsequently freed from hard manual labor and brought to modern-day Palestine/Israel by Moses and his successors. This is not reality but a myth. Archaelogical sources for the Canaanite region show a highly differantiated picture that does contradict the biblical contrast of invading, semi-nomadic, active Hebrews and pastoral, sedentary, passive Canaanites. The ethnogenesis of the "Jews" is in itself a highly complex process, that cannot be explained by the invasion of "the Jews", fled from Egypt, claiming new land as god-given. Since your question does not seem to be aimed at the ethnogenesis of the Jews I will refrain from going into more details here. The point I was trying to make is that the biblical picture is indeed that - a picture, not a photo. There were no "Jews" at the time of a postulated Exodus, whether with a historical Moses or not. The ethnogenesis of the Jewish nation happened after the "Exodus", in a complex process of both conflictual and peaceful mixture between new immigrants and the already present population, also differantiated in itself. We do not have exact record of this time-period (the transition from the Late Bronze Age to the Early Iron Age in Israel), because there are no or very few written sources, but are limited to our interpretation of archaelogical findings. These however show us quite clearly that the biblical description is inaccurate, that there was no "invasion", but that this phase of history is best described as a transition, that originated in a rapidly changing world, which - of course - brought conflict, but not in the linear way described in the bible. Cultural traditions from the Late Bronze Age linger on and show us that the population of the Iron Age (the people of the kingdoms of Juda and Israel) were probably no foreign invaders, but themselves originated in the already present population of the once Egyptian "province" of Canaan.

Modern-day (historical) theology is prone to accept a historical Exodus, but not in the manner of the biblical description, but as a small group, probably prisoners-of-war taken by the Egyptians as slaves - a common practice -, who fled (or was expelled) from Egypt and in the aftermath (re-?)settled in Canaan and blended into the already present population - their memories however survived and formed the most important part of the collective Jewish commemoration in the form of the tales surrounding the figure "Moses". This is not unlikely and has some support in Egyptian sources (see linked comment).

To piggyback off of this. I was under the impression that there is no historical evidence of a large Jewish population ever being in Egypt, let alone as slaves. Is there evidence that the Jews were even in Egypt?