This simply doesn't work as long as the generator is running.
A generator applies a changing magnetic field to a coil with an iron core. In an ideal generator, this magnetic field can't be influenced by the load current: If the current rises, you draw energy from the magnetic field, which is then replenished by the mechanical power source attached to it. (Otherwise the generator wouldn't work.)
The way the magnetic field is generated doesn't really matter for the (magnetics-only) operation of the generator. We could just as well generate it with another winding on the magnetic core, turning it into a simple transformer. The primary of that transformer is connected to a fixed AC voltage source, which is equivalent to a rotor rotating at constant RPM (which is the case in almost all generators).
Now that we know that a generator at fixed RPM is equivalent to a transformer with a fixed AC input voltage, we can do further analysis. An ideal transformer just steps various electrical quantities up or down: Voltage as N, current as 1/N, and impedance as N². In particular, this means that the output of our ideal transformer (and therefore generator) is just another ideal voltage source. With a non-ideal transformer, we also get a leakage inductance and copper resistance in series with the voltage source, as well as a magnetizing inductance in parallel to the latter. The fact that the magnetizing inductance is in parallel to the voltage source means that no matter what you do at the secondary of the transformer, the current through the magnetizing inductance doesn't change. The load current simply doesn't flow through it. The same applies to the generator: As long as it's spinning, the magnetizing inductance of its coils is permanently connected to an (imaginary) voltage source. The magnetic flux through the coils doesn't change with the load current, just like in a transformer.
If you want to measure the BH curve of a transformer, you first have to disconnect it from mains, otherwise the secondary current just gets transformed into an equivalent primary current and has nothing to do with the magnetic field in the transformer's core. The same applies to the generator by the analogy above: If you want to measure its BH curve, you have to stop its rotation, otherwise the current at its output just gets transformed into a mechanical force acting on its rotating shaft - and therefore it has nothing to do with the magnetic field in its core.
What you've been measuring is just a Lissajous curve of the load current vs. the integrated (phase-shifted) AC voltage. The fact that you get these loops simply means that you have reactance (i.e. capacitors) in the load. The flat tops indicate that there's a rectifier somewhere. (A PFC circuit in the load, maybe?)