Carpet underlayment should be used under carpeting. It is missing a component (absorption layer) to be a good generator enclosure insulation.
You should use a 1" thick vinyl/foam insulation with a 1lb sq ft vinyl barrier. For generator enclosures the peel and stick option really helps getting the insulation under the bent "lips" around the perimeter on the back of the panels. "Better" insulation will usually not contribute to an overall noise reduction. The air intake opening kind of sets the baseline for noise, and better/more baffling is the only way to improve this noise "leak."
All insulation either acoustic or thermal, will have a thermal component. Your generator is designed to accommodate an enclosure, the air intake baffle is at the generator end so "outside air" flows over the gen side before the intake manifold. The engine is water cooled and doesn't care.
There is better insulation, but for smaller generators like yours 1" 1lb is the best compromise. You probably don't have room for a thicker product, and a heavier (2lb sq ft) barrier makes the panels very heavy and create the need for mechanical fasteners. As I mentioned above, better performing insulation typically doesn't improve overall noise reduction.
Two other options that do work are secondary isolation and a gas/water separator in the exhaust run.
Secondary isolation calls for an additional set of mounts under the generator in addition to the factory mounts.
A gas/water separator will reduce exhaust noise
Isolators need to be sized for weight properly, your generator most probably has the mounting holes placed to equally distribute the weight around the C/G, the manufacturer accounted for this option.