Exoplanet Proxima b will be an important laboratory for the search for extraterrestrial life for the decades ahead. Here, we discuss the prospects of detecting carbon dioxide at 15 μm using a spectral filtering technique with the Medium Resolution Spectrograph (MRS) mode of the Mid-Infrared Instrument (MIRI) on the James Webb Space Telescope (JWST). At superior conjunction, the planet is expected to show a contrast of up to 100 ppm with respect to the star. At a spectral resolving power of R = 1790-2640, about 100 spectral CO₂ features are visible within the 13.2-15.8 μm (3B) band, which can be combined to boost the planet atmospheric signal by a factor of 3-4, depending on the atmospheric temperature structure and CO₂ abundance. If atmospheric conditions are favorable (assuming an Earth-like atmosphere), with this new application to the cross-correlation technique, carbon dioxide can be detected within a few days of JWST observations. However, this can only be achieved if both the instrumental spectral response and the stellar spectrum can be determined to a relative precision of ≤1 10^-4 between adjacent spectral channels. Absolute flux calibration is not required, and the method is insensitive to the strong broadband variability of the host star. Precise calibration of the spectral features of the host star may only be attainable by obtaining deep observations of the system during inferior conjunction that serve as a reference. The high-pass filter spectroscopic technique with the MIRI MRS can be tested on warm Jupiters, Neptunes, and super-Earths with significantly higher planet/star contrast ratios than the Proxima system.