In my case it's an Onan genset with an Onan Elite E140H 14 HP engine.
The generator is a Pro 6000E Model EGHEB

The former owner had alligator clips on the battery cables that he
used to connect to his car battery to start it, and then clamp them to
the frame when not in use. Although that's a dead short, the charging
circuit still works.

That's not the problem though. It's putting out 16V, which I suspect
will fry the little U1 battery. Apparently Onan used two different
charging systems on these gensets. The better one had a voltage
regulator and the specs for that say that it shouldn't go over 14 V.
The other type had a simple diode that just rectified the current and
sent it to the battery. I have the latter. I talked to a local small
engine repair person and he said 16-18 V is common with that type of
charging system and the current is so low that it doesn't hurt the
battery. What say you all about that diode system and the health of
the battery. Is it worth putting a regulator on, or is the alternator
so low powered in this type that it wouldn't make a difference? Is
there a common voltage regulator that I could stick on there that
would work better?