My dad did something like this for his Honda generator, the key for sound control is vibration isolation and right angles for sound to travel around. He made his with a hinged top and one end that would fold down so he could pull out the generator to service.
Sound waves are a difference in pressure in a medium at a given frequency. To control that, you need to make an enclosure that will resist (diffuse) or consume (absorb) the pressure difference at the required frequency. Higher frequencies are easier to absorb and don't travel well (think intake exhaust valves opening and closing) - unlike lower frequencies which travel well and require more design effort and mass to control (think vibration here).
If you have enough air moving through an enclosure to keep the generator cool, you should also have enough air flow to keep the enclosure material cool. For absorbing high frequencies and heat protection, I'd recommend mineral fiber lining the enclosure around the generator.
...Think about an ASK enclosure, its metal, but the insulation material might still be somewhat flammable if it has any coloring or coating on it. You could build your enclosure and cover the inside with a fire resistant material like hardi-board, cement board or even fire rated drywall...
Mineral fiber is a Class-A fire retardant material that is used for suppressing noise in things like elevator shafts - it's usually rated north of 1000C, so if it starts burning, your battery and fuel tank will have gone long before then (i.e. you'd have bigger problems to worry about). Air movement is a very powerful cooling method, using a blower that is powered from the generator head -
something like this direct-drive blower, which moves about 500-1000CFM and draws less than a 500Watt halogen flood light should do it. Keep in mind the direction of the air flow in the enclosure - your generator head has a blower in it that has an airflow preference, don't blow backwards or the generator will overheat. If you mount the blower in the same enclosure space as the generator and have it suck air from outside into the generator space, then blow out the hot air it should keep your wiring short and most of the sound contained.
Sound likes to find leaks, so make the intake or exhaust air side one of your doors so that you have less to seal. Also air speed is a generator of sound - over 1-meter/second and you start getting wind noise. Everything is interconnected