Fermi had been doing experiments with neutrons from the very beginning, with a first paper in Nature in 1934 (a Letter to the Editor on "Radioactivity Induced by Neutron Bombardment" in the May 19, 1934 issue) after Chadwick's paper describing the neutron coming in 1932. The June 16, 1934 issue of Nature had another paper by Fermi on "Possible Production of Elements of Atomic Number Higher than 92" focusing on what elements might be produced by neutron absorption in uranium. Many more experiments followed, with some interruptions as Fermi fled Italy to end up at Columbia University.
But, to address your fundamental question on what Fermi might have known about large assemblies of uranium, I will point to a 1939 paper in Physical Review, "Neutron Production and Absorption in Uranium", H. L. Anderson, E. Fermi and Leo Szilard, Physical Review 56 284-286 (August 1, 1939). The paper begins:
It has been found that there is an abundant emission of neutrons from uranium under the action of slow neutrons, and it is of interest to ascertain whether and to what extent the number of neutrons emitted exceeds the number absorbed.
Figure 1 in the paper shows the experiment:
The paper notes that "To obtain an effect of sufficient magnitude, about 200 kg of $U_{3}O_{8}$ was used" - this was not a small experiment (also note the 10 cm bar in the figure). Also, this is clearly naturally-occurring uranium, with no isotope separation (the first separation of U235 were done in December 1942 by Lawrence's lab on his cyclotron).
After discussion their measurements of neutron production (using activation of Mn present as manganese sulfate in the water), they go on to state:
From this result we may conclude that a
nuclear chain reaction could be maintained in a
system in which neutrons are slowed down without
much absorption until they reach thermal
energies and are then mostly absorbed by uranium
rather than by another element.
On of the 'other elements' of concern is hydrogen, which happily grabs thermal neutrons to make deuterium (where "happy" is about 1 in 100 neutron scattering events). Absorption and thermalization characteristics of many elements were studied previously by Fermi and others, so their general properties were well known.
So, in many ways, the Chicago Pile was a fairly straightforward engineering problem. Yes, going from 440 pounds to 80,000 pounds of uranium oxide seems like a large leap, but it actually makes much of the neutron economy math easier since the boundary effects vs volume effects are smaller. I think it is easy to say that Fermi, with long experimental and theoretical background in neutrons in uranium, had a really good idea of how the Chicago Pile would perform.
