Battlecry effects typically do not count for summon interactions. If you summon a minion that gains taunt with a battlecry (dragon priest) they didn't count towards the summon taunt minions quests. TheShellshifter may be an exception because the choose 1 mechanic isn't a battlecry so it might resolve before the creature is summoned but I had the beast quest yesterday and my Druid of the Claw and Druid of the Fang did not count towards that quest unless they were bounced to my hand after the transformation.
The difference is that the beat Hearthstone quest says "play" not "summon", while the Quest card says "summon". This is why I think it will work. Battlecry, I believe, happens BEFORE the card is actually summoned, which is why Sacred Trial used to kill your Dr. Boom when you had one minion on board. The bots hit the board first, and then he did (even though the animation looks like the opposite) and so he was the one that got axed.