In a premixed dual-fuel (DF) methane-diesel engine, the ignition of the lean premixed methane/air mixture starts with the assistance of a pilot diesel injection. Auto-ignition of pilot fuel is important as it triggers the subsequent combustion processes. A delay in the auto-ignition process may lead to misfiring, incomplete combustion, and thus higher greenhouse emissions due to methane slip. Hence, a better understanding of the auto-ignition process for the pilot fuel can help to improve the overall engine performance, combustion efficiency, and to lower exhaust emission levels. In the present study, large eddy simulation (LES) is used to investigate the auto-ignition process of micro-pilot diesel in premixed DF combustion in a constant volume combustion chamber (CVCC). The entire DF combustion processes including methane gas injection, methane/air mixing, pilot diesel injection, and ignition are simulated. The numerical model is validated against experimental data. The present numerical model is able to capture the ignition delay time (IDT) within a maximum relative difference of 7% to the measurements. A higher relative difference of 38% is obtained when methane gas injection and mixing are omitted in the simulation and the methane/air is assumed homogeneous. This demonstrates the importance of inhomogeneity pockets. To study the effects of temperature and methane inhomogeneities separately, different idealized inhomogeneities in temperature and methane distribution are considered inside the CVCC. The inhomogeneity in the temperature is observed to have a more profound influence on the IDT than the methane inhomogeneity. The inhomogeneity pockets of temperature advance the first-stage ignition and, subsequently, the second-stage ignition. A sensitivity analysis on the effect of inhomogeneity wavelength reveals that the larger wavelengths enhance the combustion due to the presence of pilot diesel jets in the desirable regions for a longer time duration.