Priest is hilariously overpowered for this brawl. Of the three runs I've done so far with the class, all three skipped the third phase entirely, once because I destroyed the Head with Convincing Infiltrator, the other two because I STOLE it with Sylvanas, allowing me to hit face or let the fireballs do it. Instant victory.
The trick is to build HARD around Lazul's absolutely broken hero power. It can produce some insane amounts of value every single turn for one mana, and can get you a board that grows completely out of control very, very quickly.
Magic Mirror comes up in actual card buckets and is a great pick when it does. Then you can use it and the Amulet of Domination you start with to go shopping for minions among the earlier bosses, or even on strong minions in your own deck you want to have more chances to draw with Hero Power. Priority obviously goes to minions with strong deathrattles, like Sylvanas (run by the normal jar I believe and also appears in card buckets), Eager Underling (7th boss), or good old Convincing Infiltrator. Reborn minions with deathrattle like Grandmummy or Khartut Defender are also not bad to have, and a lot of the bosses run them.
Some good passives I've picked up include stargazing (double the broken strategy for 0 cost? Yes please!) and Rocket Backpacks (allows you to trigger your deathrattles the turn you play them, VERY good). I haven't seen the double deathrattle totem yet (only the double battlecry one, which isn't bad either), but if it is in the run, TAKE IT! As for treasures, Ancient Reflections is good, Elder Taggawag is insane value eventually, but it can slow you down a bit, Book of the Dead is a good pick due to all the chump change dying, etc. You'll know what to do.
Without straining myself too hard, and the occasional misplay, I easily went to 33 minutes with this strategy.