Ultimately, yes you will need to bias the gate. If you put the MOSFET in the diode connection mode, the gate will automatically be biased when you force a constant Ids current. While I have never worked with this MOSFET nor with Cs137, I don’t see why it wouldn’t be sensitive. A few notes:
This MOSFET is in a TO-247 package, so make sure that you have the front of the MOSFET pointed towards the Cs137 source during irradiation, otherwise the leadframe will likely act as an attenuator, reducing the sensitivity.
This MOSFET is a HEXFET, which normally aren’t designed for continuous high DC power dissipation (they are meant for switching). So keeping the dissipated power in the FET would be best.
I’m not sure if there is a difference in general sensitivity between HEXFETs and other MOSFET types like VDMOS or traditional monolithic planar MOSFETs.
I’m not sure if you already planned this, put generally it is recommended for the MOSFET to have all pins grounded during irradiation. Biasing of the MOSFET can affect its sensitivity (depends on every MOSFET), so having all pins grounded keeps them in a constant state during irradiation (and lets all accumulated charge get shunted to ground, preventing ESD damage).