Try this and see if it works.
Instead of Abandoning the Queue, accept the queue, and as soon as you are teleported to the Raid or Dungeon, teleport back out. Re-engage the pet and then teleport back to the instance.
I could swear i did that during raid finder, when I engaged a rare oily slimeling. I accepted the queue, then teleported back out and he was still there (and i nabbed him)