There are two ways you can do this achievement:
Cheese the Challenges IV and V using a low power level character:
If you're struggling with this achievement as much as I was at a low power level, letting the other heroes do the work. It doesn't matter how many times your AI heroes get knocked out; you won't lose. You only lose if you get knocked out three times. If you're struggling to finish, try picking up a few gear pieces.
Sure, it may take longer, but I just completed challenges IV and V this way with Captain America. If you're impatient, you can pick your spots, or even use your heroic abilities. That's what I did!
Sure, we all know enemy power levels stay with your power level. However, if you don't mind waiting, in the Harm Challenge Room, enemy power levels max out at 100. If you take in an overlevelled hero it shouldn't be that hard. I tested this by taking Ms. Marvel (hero level 50, power level 119) into Challenge V, and while I struggled in certain parts (I'm bad at playing defense), I wasn't downed once. Now she's power level 134, and they're still power level 100.