In this chapter, Catwoman ups the ante by trying to steal two fancy briefcases from Hugo Strange's secret vault, just so she can annoy Poison Ivy.
Make your way towards the objective marker, where you will see a manhole guarded by the new elite enemies: TYGER Guards. One is a regular guard, one has a riot shield, and the third is armoured. Even though I engage everyone head-on in the video, that was a lucky fight. Your best bet is to take out the riot shield guard first from above, because Catwoman is weakest against those enemies, then deal with the other two separately. Once the fight is over, head into the manhole.
Keep running forward, past the Collapsed Corridor. When you reach the Confiscated Goods Vault, turn left and head through the door.
Confiscated Goods Vault
Immediately climb up to the ceiling and crawl forward. In Catwoman's stealth sections, you cannot afford to be seen for more than a few seconds, no matter what the situation. You need to pickpocket three cards from the TYGER guards to open the vault. Pickpocketing works the same way as Silent Takedowns. Once you've pickpocketed enough keys, head back to the computer to open the vault.
After the vault is opened you'll need to take out the guards. Follow the route I took in the video; there's a lot of waiting around, but it's easy and worth it in the end. I also messed up a bit at the end, which will show you what you should do in the event you do get spotted.
Once everyone's down, go through the vault door. Smash Ivy's plant and take the briefcases with Catwoman's loot. This triggers another group fight with a Riot Shield guard, four regular guards, and an armoured guard. Use your special takedown combo to get rid of the riot shield guard, and then you can proceed to beat up the other guards as normal.
Once they're down, grab the briefcases and walk out the loot door. You are then presented with a choice to either rescue Batman or leave with the loot. Doing the latter will trigger a non-standard game over, so drop the briefcases when prompted, and run back to the sewer. The tunnel will collapse behind you. Climb up the ladder.