Hair is protein, mostly keratin. The hair color chemistry is made to bind to that.
Wood is mostly cellulose, a carbohydrate that very few critters can digest at all.
Wood dyes bind to cellulose, not protein with the same effect.
Find some post secondary botany lab. See if you can score a gram or two of Saffranin O.
Dissolve that in maybe 500ml Everclear (or 95% if you can find it.)
Don't use denatured lab alcohol. It's usually poisoned with some crap like Jet B that spoils the binding.
Test on scrap for time. I'll predict soaking for an hour+.
Saffranin O in alcohol should stick to your skin for less than a month.