A few further points:
These engines are noisy by nature because the finned cylinders and heads must have direct access to cooling air.
If your generator has seen only occasional use, (not worn and loose enough that it's using oil), a short-block isn't going to change things. If you do go in that direction, be aware that not any old short-block will do. Generators have a special crankshaft with a long taper, rather than a key on the drive-end -- the long center bolt draws the rotor's seat onto the taper.
I imagine your Coleman used the standard cube-shaped two-bolt Briggs muffler. Better-quality generators, for example, some older Homelite HG's with Briggs I/C's, were equipped with tuned mufflers the size of a small telephone book. These mufflers are furnished by the gennie manufacturer, not Briggs, and do noticeably reduce noise. Because all that weight on the port flange and fasteners acts as a bob-weight, the mufflers are always braced to the generator's end casting or its mounting feet. You might ask small engine service people for more current suggestions.
Generator mounting cushions are also tuned to selectively isolate vibration. Don't substitute with generic shock mounts from Grainger's. (Too rigid -- I've seen two cases where this caused the crankshaft's ball bearing race to wear its mating bore in the aluminum case. Adds a squealing noise to the mix.) Unless crushed-down, cracked or oil-soaked, there's probably nothing to be gained by replacing them.
One point that hasn't been mentioned in connection with enclosures is that fumes from refueling and the unavoidable spillage dissipate before the gennie can be safely fired up. If it's not brushless, there will be opportunities for sparks, aside from the fact that hot exhaust parts can be sources of ignition.
All things considered, it's not worth spending big $$$ on a ten year-old homeowner-quality generator, so try low-buck approaches first. If that doesn't work out and you're thinking about a new unit, the quiet, very expensive inverter models only run as fast as the load dictates. Auto-idle, which reduces speed under no-load conditions, has long been available on commercial-quality units because it reduces fuel consumption. Anything other than a homeowner-quality generator should have overhead valves and a cast-iron cylinder bore for longer life. Honda, Kawasaki, Fuji-Robin, Yamaha, and Briggs Vanguard series are good, and you won't be able to find parts for unknown Brand-X Chinese clones, which also cheat on the business end. You might talk to the people at contractor rental outfits for recommendations.