I haven't looked at the Anvil in any detail. However, your question is fortuitous as I think I learned something useful yesterday. I wrapped tin platted copper wire around the cap. This is that heat-battery noise going on. What I learned is that the wire seems to limit the induction to the cap significantly. I got loads of hits but never reached dark plant matter.
This means that copper and/or the thin tin plate on the wire is not "RF transparent". It shielded the induction to the cap and probably had a weak reaction to copper. I did not know I could shield the cap from induction heating like like that. Now I need to find out if silver heats but I don't hold a lot of hope for that. Silver is much like copper, highly conductive. My bet is on music wire. That I might test in my setup.
The other aspect is that induction is a skin-effect. It doesn't go deep. The heat radiates but is very intense where the outside skin is. Anything inside the cap is shielded from induced currents by the cap itself, even the inside surface of the cap. So the geometry of the Anvil must have enough exposed magnetic series stainless steel to provide heat. How much stainless steel surface is inducing current from the IH is not difficult to ascertain with a small lab power supply and an Anvil in hand.
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