I had a nice array of types to deal with all the sub-types, so I was able to send out a fresh Pokémon for each battle and had absolutely no problems. I guess my team wasn't designed for the Ironworks Chamber. I had a hard time finding weaknesses to exploit, and the last Pokémon might have wrecked me if I didn't already have him over-leveled. Once I got past there, I healed up, checked my supplies (I used Leppa Berries on Azumarill's Surf because I had to keep Rain Dance there), and knew I was fine from then on.
Interestingly, I faced Drasna (Dragonmark Chamber) at the Battle Chateau today.