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Can the steel of a ferromagnetic transformer instead be fashioned as a set of windings, acting as a secondary?
Here I am building prototype transformers for testing, using 1 ft length aluminum conductors of 1/2 by 1/2 inch width square conductors. I am structuring the (aluminum square tube)windings(connected at corners by ordinary round copper wire) in return layer diagonals, ( the pattern set by odd magic square sequences) to establish maximum internal capacity in the windings. If the current in the windings is opposite between every internal lateral square facing, most of the external field around the windings is cancelled, but internally between the windings the magnetic loops will be in unison. I therefore wish to put insulated steel tube sleeves around the internal aluminum loops of the wiring array, for a very unconventional honeycomb type core transformer. I should be able to measure for eddy current output on any steel sleeve by shorting the ends of the outside steel segment with an amperage meter. A conv. secondary will also need copper loops wound lengthwise around the steel.
An AC transformer connects AmpereTurns in the Primary to AmpereTurns in the Secondary using the time varying magnetic field in the core. The maths for the size of the core is well established.
If you double back on turns they do not count as part of your ampereturns - unless you shield the doubled bit. Such methods are used in shaded pole motors.
The only way to lighten the Tx is to use higher frequencies in the core. This 'dumps' the magnetic energy into the secondary more often so you can use a smaller core (bucket of energy!)
The magnetic field is always perpendicular to the current and hence so are the core loops (in a Vector Notation sense - not necessarily physically)
If you introduce Iron windings in the windings space you will get bigger Ohmic Losses.
I hope this helps - you obviously have a plan to make hyper-efficient Tx but they are already near 0.999 efficiency.
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