A lean azimuthal flame (LEAF) combustor fuelled with Jet-A1 is numerically simulated. The LEAF concept, motivated by the ``flameless'' oxidation principle, is based on azimuthally arranged fuel sprays and opposed injections of air in an axisymmetric chamber with a central outlet. This unique mixing configuration results in a highly turbulent toroidal reaction zone and achieves ultra-low emissions, making it attractive for future aero-engines. In this work, Incompletely Stirred Reactor (ISR) theory is employed to assess its capability in predicting the experimentally measured NOx concentration and soot particle size distribution (PSD) at the exhaust. Detailed fuel pyrolysis and oxidation chemistry, NOx formation models, and a detailed physicochemical sectional soot model are employed and solved in a post-processing fashion, leveraging a RANS/flamelet simulation with simple chemistry of a condition with thermal power of 20 kW. It is found that pollutant formation can be well reproduced in this burner, provided that an accurate energy balance is available to quantify the heat loss through the water-cooled aluminium frame and the air-cooled quartz windows present in the experiments. A parametric analysis showed that a departure from the adiabatic condition by even a few hundred W could substantially improve the agreement with measurements. In particular, heat losses were found to be responsible for a significant reduction in NOx emission, affected primarily by NNH and N2O routes, and an increase in soot emission at the burner exit, influenced mainly by finite-rate soot oxidation. Equivalently, it may be concluded that using realistic walls in the experiments, such as ceramic coatings that better reflect aviation gas turbines, would result in lower soot emission but higher NO emission. In this direction, the ISR approach and network approaches based on ISRs may be employed for preliminary evaluations and to identify suitable mixing patterns and residence times that ensure low pollutant formation across a wide operating range.