Well...
If the chainsaw is to run slow with not much power to be quiet, then maybe a weed whacker motor could be used instead. They are fairly quiet, and could be run more in their power range (with reduction). You may also have a problem with the clutch on the chainsaw. I don't know if you intend to bypass it (or just chuck it and put the pulley right on the crank) , but if you leave the clutch in place, it woldn't be able to engage promptly as it was designed to and would quickly burn out.
Another thought I had (which I have done myself, and with good results) is to find a Tecumseh 2 stroke from a single stage snowblower. The muffler that comes bolted to the engine usually isn't too quiet, but they made a manafold/header type of thing that would go on in its place on some snow blowers. It would have a muffler that was usually stuck between the two wheels and acted as a frame member. It would be out of the question to use that muffler as it is huge. What I would suggest is getting the header, piece of pipe that lead to the muffler in the frame and make your own muffler contraption. Get the pipe to say put in the header, then stick the threaded ends of a Briggs and Stratton style muffler in to the pipe. It will slide in and fit nicely. Braze/weld/fasten it in place. Weld another Briggs muffler to the end of the one in the tube, and maybe another one to that. You would end up with three or so inline. I have my bike done up this way, but only one muffler is used as that is all I really wanted. I have put more than one muffler up to it and it turns to about the sound of a common Chinese bike motor, or about that of a common weed whacker or less.
The Tecumseh motor is about three horsepower. Mine can do 35 or so before it is really screaming and I start to fear for my life. The set up has another advantage of the Tecumseh having a common shaft diameter so you have a wide range of pulley options, mine is 5/8". You could have a belt drive to the jackshaft, and use a tensioner as a clutch. You could still go with the sprocket idea and have a sprocket go from the jackshaft to the back wheel. My bike is a 26 inch, so you wouldn't even need as much as mine had. I would imagine a 2 or 3inch pulley to a 6 or 8 inch on the jackshaft with a 10-15 tooth on the jackshaft to a common crank sprocket (about 30 teeth) would be plenty enough. Calculated out, this would give you about 18 miles an hour with the engine running at 3600 RPM. Using the bigger sizes for the driving pieces (pulley on engine and sprocket on jackshaft) or smaller driven pieces (pulley on jackshaft, sprocket on wheel) will get you higher speeds, and just the opposite for lower speeds.
My bike is geared a little higher than that for more speed, but I sort of had to do that because of the large tire size and price of large sprockets and pulleys. As a side note, try to find an engine with adjustment screws in the carb, the emissions (screwless) ones SUCK!
Good day,
Brian