So, a brief bit of explanation. The part that needs to be replaced is an ignition rotor. It is keyed to the crankshaft, and strategically located at the near exact mid-point of this machine... right between the engine section, and the rotor/stator. It costs about $15.
To get at it is a bit of a puzzle however, and it starts with getting the thing out of a small generator shed on the end of the woodshed, and into the shop. Briefly:
Turn off the gas (LPG),
disconnect the remote wiring harness,
disconnect the battery,
disconnect the power leads,
disconnect the muffler,
disconnect the gas line,
unbolt the base,
carefully lift above said bolts,
carefully load into trailer,
move to shop

Once there, we begin another sequence:
Drain the oil,
remove the brushes,
remove the cowlings, covers,
remove the air filter housing & carb assy.,
remove the PTO coupling,
remove the retainer and thru-bolt,
remove the base plate
Now it can be tilted onto the blower housing. It is a tapered-fit, press fit which must be jacked apart. A small 'parting tool' - mine is a piece of threaded rod with saw kerf in one end - is threaded in, and a larger (1/2" IIRC) bolt is torqued against the end to pop them apart.
Carefully lift, and separate the two halves... There is the part we're after. A small keyed piece of plastic with a pencil eraser-sized magnet in each end. When both halves are there, once per rev, the ignition switches on, and fires the plugs.

The location I suspect is tempting for a mouse. It sits above the oil reservoir, which stays warm long after the engine has been shut off on a cold fall day. I'm not sure how to keep them out; clearly. The entry is where the starter engages the flywheel. It's very small, but there doesn't seem to be a viable way screen it off. Making the shed impervious may be a better option, but sliding door could be a challenge. I'm open to ideas.
The last time this happened was 2007. I bought two of them back then. Maybe that should have been an omen. My parts supplier, who does RV's said then, "yup, this happens all the time with motor homes (and this generator)."