I think 3 theorists have Solid Iron Sun models: Brant Callahan; Michael Mozina; and Oliver Manuel. I think Charles Chandler and Wal Thornhill consider the Sun to have considerable iron content, but below .7 solar radius.
There seem to be two main problems with the 3 solid iron Sun theories.
1. How can the Sun have a solid iron shell, whether 500 or 5,000 km below the top of the photosphere, and still have differential rotation at different latitudes?
2. How can a solid iron shell maintain charge separation that allows electric discharges to arc from the iron surface into the solar atmosphere and then back down to the solid iron shell?