The Cutout wont work unless its well grounded, try a jumper wire from its metal case to a known good clean frame ground......
If she charges strong if you by pass jump across the cutout (wire BAT to GEN/ARM) that tells me the genny is okay and the problem is the relay isnt latching in cuz its just BAD or its not grounded.
Is it the correct polarity to match the tractor i.e. Pos or Neg ground???
Run through my Troubleshooting Procedure