1. The Genset itself is NOT the problem
2. The Genset has NO charging capability. THAT is done thru the Power Converter or Inverter/Charger which ever you have.
3. I would suspect the Battery Positive cable between the fuse and the connection at the genset is melted/crimped/shorted. OR routed close to the Genset exhaust OR Engine Exhaust(I pick the Engine exhaust). That is why the fuse is blowing after running and the Geset and Engine exhaust gets HOT. Good thing the OEM installed the that fuse or you would have a good fire on a shorted 12 volt Positive cable. Doug