We present irradiation experiments carried out on RadFETs in the high-intensity T2 neutron beam at the CRC-UCL in Belgium. The aim of the test was to characterize the neutron response of RadFETs in view of their use as an integrated part of a radiation-monitoring sensor for the CERN Large Hadron Collider (LHC) experiments. Two types of RadFETs were investigated up to a total neutron fluence of corresponding to a deposited dose of 744 Gy in silicon. The responses of bare devices to neutrons are compared to the commonly used reference measurements with gamma rays. It is found that the gamma ray calibration cannot directly be adopted to convert the RadFET signals into neutron dose. In a second experiment, the influence of a plastic packaging, simulated by polyethylene slabs of different thicknesses, was tested in the neutron beam and compared to GEANT4 Monte Carlo simulations. An increase of the RadFET neutron sensitivity by a factor of up to 7 due to the packaging is found. The influence of these findings on the conception of the radiation-monitoring sensor is discussed.