We report the time-resolved spectral analysis of a bright near-infrared and moderate X-ray flare of Sgr A*. We obtained light curves in the $M$-, $K$-, and $H$-bands in the mid- and near-infrared and in the $2-8~mathrm{keV}$ and $2-70~mathrm{keV}$ bands in the X-ray. The observed spectral slope in the near-infrared band is $ u L_ upropto u^{0.5pm0.2}$; the spectral slope observed in the X-ray band is $ u L_ u propto u^{-0.7pm0.5}$. We tested synchrotron and synchrotron self-Compton (SSC) scenarios. The observed near-infrared brightness and X-ray faintness, together with the observed spectral slopes, pose challenges for all models explored. We rule out a scenario in which the near-infrared emission is synchrotron emission and the X-ray emission is SSC. A one-zone model in which both the near-infrared and X-ray luminosity are produced by SSC and a model in which the luminosity stems from a cooled synchrotron spectrum can explain the flare. In order to describe the mean SED, both models require specific values of the maximum Lorentz factor $gamma_{max}$, which however differ by roughly two orders of magnitude: the SSC model suggests that electrons are accelerated to $gamma_{max}sim 500$, while cooled synchrotron model requires acceleration up to $gamma_{max}sim5times 10^{4}$. The SSC scenario requires electron densities of $10^{10}~mathrm{cm^{-3}}$ much larger than typical ambient densities in the accretion flow, and thus require in an extraordinary accretion event. In contrast, assuming a source size of $1R_s$, the cooled synchrotron scenario can be realized with densities and magnetic fields comparable with the ambient accretion flow. For both models, the temporal evolution is regulated through the maximum acceleration factor $gamma_{max}$, implying that sustained particle acceleration is required to explain at least a part of the temporal evolution of the flare.
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