If there is a blocking diode in the charging circuit, check that it is OK and not short-circuit. It sounds as though the battery is running back into the DC start windings when the generator is stopped.
This function may also be carried out by a relay that has the contacts welded together.
On our SOM generator, we fitted a blocking diode as we didn't have the full set of control boxes to carry out that function, seen on the right of the box with two wires:

There is about 36 volts available for battery charging at full power, so volt drop isn't an issue.
You can also replicate the two-level charge rate by switching different taps on that resistor, which is effectively in series between the generator DC output and the battery.
Peter