I am thinking that direction too. Simply redirecting the sound, vs. enclosing the genny entirely, would help with the air flow and heat issues.
From what I have read, putting a muffler on a generator isn't much worth the trouble, because so much of the generator's noise is from the motor itself, not the exhaust.
While it may appear somewhat arcane to most of us (me anyway, most of the time), proper acoustic engineering can accomplish quite a bit. The other thing I meant to mention was check out Generac standby enclosures - around here Home Depot used to have one mocked up in-store - a good friend of mine has one that runs on NatGas, it is impressively quiet, esp. with the lid closed.
Air cooled engines do tend to be inherently noisy - one of the reasons the original Bug was noisy for it's size. And... even as air-cooled engines go, not much engineering time goes into block-radiated noise with cheap genny engines.
That said, until you get to Honda, the "mufflers" on most small generators aren't much more than spark arrestors.
Design-wise, the cost/benefit goes something like "Cheap out on the spark arrestor - get sued by a major State when one of their Parks goes up in flames. Cheap out on a "muffler" - good luck to Joe Blow for suing us for hearing damage".
Dug back till I found the link I was talking about. Got a few details wrong (Gen type, etc.... ), but general concept was correct. Yes, you won't get Honda quiet with just a muffler, but the OP guesstimates 50% reduction and in the Comments one guy meaured "103 db of sound at the muffler of the original generator stock/ now 92 db noise at the muffler area of the modified generator currently" - using a similar bike muffler setup.
You have to read down, but the pictures don't make it clear that he is NOT supporting the weight of the new muffler with
just the OE exhaust studs. He did fab up a relevant new support.
Joel's Garage: Quiet Generator Muffler
In looking back for that one today, I came across this one:
Quieten That Generator
Good history of what he did with mufflers. I like the welded flange + flex- hose approach, as it allows you to run the OE setup if you are in hurry or don't care about the noise, but allows the add-on muffler to be quickly attached. Obviously, fire safety would dictate appropriate attention/vigilance is needed when using an add-on like that.
Rgds, D.