Are there any fundamental violations in my reasoning?
Yes, conservation of energy.
The energy gained from reabsorbing radiation can't be higher than the energy lost to radiation.
To address your edit: If you have some power (say 1000 W) coming in, you will reach thermal equilibrium when radiative heat losses equal 1000W.
By putting a reflective coating and recouping 99% of losses, then the wire will heat up to whatever temperature allows it to radiate at $\frac{1000}{0.99} = 100,000$W, so that the supplied 1000W keep it in equilibrium.
Of course the more efficient the reflective coating is, the more you have to worry about the coating itself being able to withstand the head and not radiate heat itself.