A side note first - when dealing with non-physicists, they will generally regard quantum mechanics as the end-all-be-all of physics, the coolest weirdest stuff. So, it is not surprising that your colleague focused on quantum mechanics and the rest was just engineering.
So, lets look at your list first, then go on to some others of note that worked in the Manhattan Project:
Enrico Fermi: His Nobel Prize, awarded in 1938, was “for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons”. Well, that seems pretty core to the Project given the need to know deeply about nuclear reactions, neutron-driven reactions, and, of course, the production of large amounts of Plutonium through neutron irradiation needed for Fat Man. Note that Plutonium was only discovered in 1940. (Your list does not have Glenn Seaborg on it, who produced and isolated the first samples of Plutonium in February, 1941. This lead to his Nobel Prize in Chemistry in 1951, as well as having an element named after him, an honor with a higher distinction than a Nobel Prize). Fermi led the design of the Chicago Pile, which you may consider engineering but required deep knowledge of nuclear physics.
Hans Bethe - While his Nobel Prize, “for his contributions to the theory of nuclear reactions, especially his discoveries concerning the energy production in stars”, did not come until 1967, it traces back to his 1939 paper "Energy Production in Stars", Physical Review 55, 434-456 (1939). Of note is the first sentence in the introduction: 'The progress of nuclear physics in the last few years makes it possible to decide rather definitely which processes can and which cannot occur in the interior of stars." It is all about analyzing nuclear reaction chains, which is a skill quite useful in the Project.
Robert Oppenheimer - perhaps the closest related to quantum mechanics per se, given he is part of the Born-Oppenheimer approach for molecular wave functions. However he is also known for the Oppenheimer-Phillips process that allows adding protons to nuclei using deuterium, lowering the Coulomb barrier. Pre-war he was deeply involved in the nuclear physics activities at Berkeley (more below). His role on the project was as the head of it, so his contributions were more managerial, not technical.
Ernest Lawrence: His 1939 Nobel Prize was “for the invention and development of the cyclotron and for results obtained with it, especially with regard to artificial radioactive elements”. The cyclotron which he invented and made practical was a key experimental apparatus in making and understanding nuclear reactions, leading to understanding of nuclear structure and early nuclear physics. His experience with accelerators also led to the use of the Calutrons at Oak Ridge for uranium isotope separation for Little Boy.
Richard Feynman - Note that he was a freshly minted PhD who ended up in Los Alamos when basically the entire group of Wheeler was moved there. He worked in the computational group, overseeing the IBM punch card machines. Certainly, being there with all the other names probably helped him come up to speed as a professional. His Nobel Prize in 1965 was “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”. Given the nuclear physics going on at Los Alamos, one is tempted to draw connections between the Nobel Prize and the Project.
Eugene Wigner: Another physicist with ties to both quantum and nuclear physics. However, his 1963 Nobel Prize was “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”. So, while I "know" him far better for solid state physics, he was recognized more for his nuclear physics contributions. Wigner led the design of the nuclear reactors to produce plutonium, the first reactors constructed. Doctoral students include Victor Weisskopf (nuclear physics) and John Bardeen (only two-time Nobel Prize winner in Physics). Wigner was also present at the meeting between Leo Szilard and Albert Einstein that lead to the letter to Roosevelt that started the Project.
Leo Szilard - a quite wide ranging polymath, he conceived the idea of a nuclear chain reaction in 1933, shortly after the discover of the neutron. He worked with Fermi early in the Project on the Chicago Pile, then moved into reactor design. Post-war he moved into biology, including nuclear medicine.
Lets look at some other Nobel Prize winners, pre-war.
Werner Heisenberg, 1932, “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen” - he was a leader of the German program.
Paul Dirac, 1933, “for the discovery of new productive forms of atomic theory” - during the war he focused on theory and experiments of gas centrifuges for isotope separation. (his co-winnder, Schrodinger, was in Dublin through the war but remained an Austrian citizen until 1948. Thus Schrodinger did not participate in the Project).
James Chadwick, 1935, “for the discovery of the neutron”. He was deeply involved with nuclear physics, first under Rutherford and then in his own right at Liverpool. Interned in Germany during the First World War. He was closely associated with Britain's Tube Alloys project, and followed it to Los Alamos. Initial work in Britain included measuring the cross section of U-235, critical to understanding the chain reaction. Chadwick was the head of the British Mission, that oversaw all joint technical work.
Carl Anderson, 1936, “for his discovery of the positron”. During the war he did research in rocketry, was not associated with the Project. Anderson's co-winner, Victor Hess, was not involved in the Project either.
Clinton Davisson, 1937, “for their experimental discovery of the diffraction of electrons by crystals”, did not take part in the project. In contrast, the co-winner George Thomson, had gone into nuclear physiscs. Thomson was the chair of the MAUD committee that determined that an atmoic bomb was feasible.
So, the last 'quantum mechanics' Nobel Prize was awarded in 1933. After that, nuclear physics was coming into being. The confluence of World War 2 and the realization that an atomic weapon was conceivable resulted in a huge physics and engineering effort. It took a huge, concentrated effort to develop theory, mature it, and then implement the theory in engineering. It took a similar effort on the experimental side.
As a final note, you casually toss out cross-section measurements as if they are not very important. They are the core of understanding nuclear physics since they are the only way to probe nuclear energy levels and states. In this context it is important to note that John Cockroft and Ernest Walton (with Mark Oliphant) performed the first human-controlled nuclear reaction in 1932, measuring the energy dependence of the Li7(p,$\alpha$)$\alpha$ reaction. Without these measurements, nuclear physics wasn't going much of anywhere (and their measurement of the energy released is within 1% of the currently accepted Q value - not bad for the first ever measurement). Cockroft was also part of the MAUD Committee, but went into radar work for the war. Cockroft and Walton won the 1951 Nobel Prize “for their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles”.
So, I would agree that 'quantum mechanics' such as an undergraduate takes was not necessary at all for the Manhattan Project. However, in the 1930s, quantum mechanics generally well established, and was rapidly evolving into what we now call nuclear and particle physics. Most of the names you mention are better known for these later efforts. The fact that 1932 saw the discovery of the neutron, the first human-controlled splitting of an atom, and by 1933 the realization that a chain reaction was possible shows how quickly people were taking 'classical' quantum mechanics and applying those concepts to understanding nuclear and particle physics. During the Manhattan Project huge strides were made, experimentally and theoretically, in understanding nuclear physics and realizing nuclear physics as a weapon.
