You got it Motor.
Update- After many hours adapting the weed wacker to drive the alternator- the wacker has not enough torque to run the alt. I could not even start the thing with the belt connected.
I have a rain check for the 2.5 hp engine from HF and should be able to pick it up this friday. Only issue is the 23# it weighs! But it will have torque. I am looking at a full up weight of over 70 pounds for the Mongoose with all the drive stuff. Heavy, but no heavier than some electric bikes.
Yes, the engine will be running at all times with varying rpm. The engine and alternator will go on a rack. As for noise level, I will make a muffler and sound attenuate the deal later. It WILL be quiet.

I just ordered a sprocket and attachment to the back wheel. Real unsure about drive ratio, I have 5.2 from the electric scooter to the jackshaft (which is a rear hub from a single speed 20 inch bike with the freewheel side having the scooter sprocket and the other side as small sprocket as I can weld on), the wheel sprocket I ordered is a 44 and the sprocket I welded to the "Jackshaft" is about 2.5 " - maybe 2.5 to one. Can go bigger on the rear sprocket if needed.

I have had a bunch of stuff to do lately (winter preparation-snow tires, rustproofing wife's car with oil/cosmoline........) and will be able to do some more on the bike, especially since rain is forecast fo the next several days.
What is unique to me, is that as far as I know, no one has done this kind of set-up.
If it wasn't for the NY law making 250 watts legal- I would not have attempted it.
If the thing works out, it could be repeated in places like Austrellia (spell?).