Proxima b is a rocky exoplanet in the habitable zone of the nearest star system and a key test case in the search for extraterrestrial life. Here, we investigate the characterization of a potential Earth-like atmosphere around Proxima b in reflected light via molecule mapping, combining high-resolution spectroscopy (HRS) and high contrast imaging, using the first-generation integral field spectrograph, High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph (HARMONI), on the 39-m Extremely Large Telescope (ELT). We simulate comprehensive observations of Proxima b at an assumed 45° inclination using HARMONI's High Contrast Adaptive Optics mode, with spatial resolution $\sim 8 \,\rm mas$ ($3.88 \,\rm mas$ spaxel^-1) and spectral resolving power R ≃ 17 000 between 1.538-$1.678\,\mathrm{\mu }\mathrm{m}$, containing the spectral features of water, carbon dioxide, and methane. Tellurics, stellar features, and additional noise sources are included, and removed using established molecule mapping techniques. We find that HARMONI's current focal plane mask (FPM) is too large and obscures the orbit of Proxima b and thus explore smaller and offset FPMs to yield a detection. An $\rm {S/N}=5$ detection of Proxima b's reflected light, suitable for atmospheric characterization, is possible with such modifications, requiring a minimum of 20 h, but ideally at least 30 h of integration time. We highlight that such detections do not scale with the photon noise, hence suitably detailed simulations of future instruments for the ELTs are needed to fully understand their ability to perform HRS observations of exoplanet atmospheres. Alterations to the HARMONI FPM design are feasible at this stage, but must be considered in context of other science cases.