Silicon Photomultipliers with cell-pitch ranging from 12 $mu$m to 20 $mu$m were tested against neutron irradiation at moderate fluences to study their performance for calorimetric applications. The photosensors were developed by FBK employing the RGB-HD technology. We performed irradiation tests up to $2 times 10^{11}$ n/cm$^2$ (1 MeV eq.) at the INFN-LNL Irradiation Test facility. The SiPMs were characterized on-site (dark current and photoelectron response) during and after irradiations at different fluences. The irradiated SiPMs were installed in the ENUBET compact calorimetric modules and characterized with muons and electrons at the CERN East Area facility. The tests demonstrate that both the electromagnetic response and the sensitivity to minimum ionizing particles are retained after irradiation. Gain compensation can be achieved increasing the bias voltage well within the operation range of the SiPMs. The sensitivity to single photoelectrons is lost at $sim 10^{10}$ n/cm$^2$ due to the increase of the dark current.
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