When the circle members doubt that Cole actually saw a Spirit Bear after his first return from the island, why does Cole chose not to show them the white tuft of hair as proof?

Cole is insulted by their lack of trust. Showing the white tuft is certainly not conclusive proof of anything. To show them the tuft would be to belittle his own story. Also Cole's word should be enough. Cole is not a conformist and does not wish to appeal to people who do not believe him.