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Is there no measurable quantity in the current state of a magnetized ball that would predict its behavior in a magnetic field then?

I'd be surprised if that were the case: seems like it'd violate some sort of "locality", where the future should depend entirely on the present state.

Is it just that there's unidentified present state that's perhaps more easily characterized by knowing the history? Or is it somehow unidentifiable in principle?

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Yeah, the current magnetization of the sphere carries all the history of the magnetic fields it's been in in its value.

The hysteresis in the material only means that if you apply 1T to it to magnetise (supposing the ball starts with zero magnetisation) it and then apply 0T to demagnetise it, the remanent field in the ball would be like 0.4T due to hystetesis, you need to apply something like -0.4T to leave it as it started.

Thank you! That makes sense to me.

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