The running only after "warm" does have me scratching my head.
Now I know on some alternators the field does not kick on till a few seconds after starting. Generator maybe never had one.
OR you got something getting hot and expanding enough to touch something else and making a circuit on the coil side of things bypassing the key.
Either way I would start pulling some fuses and minimizing what needs to be HOT to run and test this out.
