Gamma-ray spectroscopy measurements at MHz counting rates have been carried out, for the first time, with a compact spectrometer based on a LaBr3 scintillator and silicon photomultipliers. The instrument, which is also insensitive to magnetic fields, has been developed in view of the upgrade of the gamma-ray camera diagnostic for α particle measurements in deuterium-tritium plasmas of the Joint European Torus. Spectra were measured up to 2.9 MHz with a projected energy resolution of 3%-4% in the 3-5 MeV range, of interest for fast ion physics studies in fusion plasmas. The results reported here pave the way to first time measurements of the confined α particle profile in high power plasmas of the next deuterium-tritium campaign at the Joint European Torus.