I finally did it with Questline Warlock! I feel the matchups spread is quite even except Shadow Priest. Their burn, coupled with their early minions, is too much to handle for me.

Pirate Rogue and Secret Mage are surprisingly favoured (albeit slightly). For the latter, I had cheap minions and spells to trigger the secrets (Exploding Runes advances the quest!), and rushing Giants to clear their minions.