Is that around the moment arm?
As to thickness. It depends. I’d recommend spring steel otherwise it will hold it’d bend if the headrest is deflected too much. Also, steel is cheap. Buy 0.5, 1.0 and 1.5mm, make them all, determine which is best.
I’d also go with stainless steel.
Do something about the form, your rectangle of steel is not good (and I’m being kind by saying not good). Also consider buying a 3.0 and 6.0mm piece of aluminum and making prototypes.
Quite frankly in a problem like this, drawings and opinions suck. Make prototypes, then decide.