I presume a C or E core (closed magnetic circuit) will not do, I suppose a soft iron rod, or a section of metglas from a C core will be ideal (far less winding with metglas) I will likely obtain metglas only if I determine winding with softiron will take substantial time and it likely will, considering 12awg is of great diameter which demands more turns for increased inductance.
There is no use in attempting to run the gizmo without some current inhibiting element when sims displayed RMS currents far surpassing breaker ratings.
Regardless of how cold MOSFETs will remain as they conduct power within the gizmo (Im assuming your reference was to MOSFETs), I rather stick with the simplicity of 1 very high current/power rated IGBT and if heat becomes an issue then paralleling them will surely resolve it. SCRs are far cheaper but at the cost of gate trigger circuitry - more parts and ultimately more things which can go wrong and cause things to really blow up. Im not sure what corners you were referencing, perhaps the price breakpoints for various part quantities?
Don't if this helps but I've used this type of method before for dumping and charging a capacitor after trying many configurations with scrs gave up and used a good mosfet.View attachment Three Cap Dump Methods 8-25-12.pdf