I think eliminating residual flux in iron core may be like degaussing of magnetic tape ie applying strong alternating magnetic field such that its peak exceeds the saturation of magnetic core and decrease it gradually

I agree with saturating the material and then demagnetising it,of course this would require a very controlled setup with accurate sensors to monitor the flux in the sample.

The curie temperature may work too but you may lose some magnetic material properties.

You should made an air gap in a core