If I play a Silumgar Butcher and, in response to its exploit trigger, it becomes a copy of Sidisi, Undead Vizier, will I get to Demonic Tutor if I sacrifice a creature?

A creature's exploit ability is not linked to its “when this exploits” triggered ability. If your Butcher becomes a Sidisi in response to the exploit ability, you will indeed get to tutor if the Butcher-Sidisi exploits something.