Say my opponent initiates a challenge against me and I have a Maester with these chains: Valyrian Steel Link , Copper Link, Pale Steel Link and Bronze Steel Link.

As a player action, I kneel to draw 2 cards then begin to trigger my responses. I use the Copper link to give the dragon trait, then use the Pale Steel Link to put Dragon Bite into play on the character attacking (provided they are a valid target naturally).

Will the character immediately die and my attachment put in to my discard pile in time for me to use the response on Bronze Link to return it to hand?

The character would die, but both the dead character and the attachment will be moribund and on the table. So Dragon Bite will not actually be in your discard pile for Bronze link to return to your hand.