I'd make it Fury warrior (he can slow. fury with 2pc t12 got 1 spare point which can be put into the slow) and a range dps. The warriors job is to slow + cleave while blocking the exploding spiders (3 of them, last one should be made by someone else, preferably the range dps).
And yes that's what I'm doing every week on my warrior. It works although it's not as much fun as hitting a mob
We have two people soaking.
Me as a Boomkin Aoeing the spiders and soaking broods, and a warlock was also soaking the spiders that i couldn't reach due to the poison on the ground.
And if someone was fixiated then could just run out and soak themself if needed.
But for you the shadowpriest is the only real option for you.
Frost DKs are great at it if your Blood DK/Fury Warrior have decent off-specs. But you can do it with a variety of classes; for your specific comp you could have one of your mages go Fire and do it, or your warrior can do it - in each case the spriest will probably be needed to help a little bit at times.
Its certainly possible to 2 heal, but for progression 3 heals is really going to help you learn the fight and will be more stable.