This is actually pretty easy once you figure out when to attack Avie when you get to the cloud. If you do don't attack at the correct moments you are sent back to the 1000th floor. I couldn't find anything that really helped aside from a video of the fight and someone mentioning you have to hit her when she wasn't attacking.
I went into the fight with Beggar Queen (hit harder, less money +2 attack). Pugilist would be another option as you are fighting Avie without weapons.
Choose 1 of the 5 available behind each shopkeep. Dodge, speed, higher attack. Whatever you can get to make you harder to hit and save your health. The only attacking I did was the first fight on top of the building, after that you only need to dodge lasers and wait for the app to drop to go back to the cloud to fight again.
Fighting Avie: when you do this, dodge until she's not glowing, a dark color, and on a knee. The first fight is easy enough. When you get sent back to the building, dodge until the app drops. Dodge attacks until she's not glowing. The window you have to attack will be shorter until she attacks again. If you attack too many times when she's not in the vulnerable state you will be sent back to the roof. As long as you keep dodge rolling and pay attention to health you should be good. In the third fight, the window to attack is maybe a couple of seconds. She attacks, will be vulnerable for maybe 2 or 3 seconds before attacking again. Hit her again and it's over.
I did have an issue after my second fight with Avie in which it took down her health, I was sent back to the roof, and her health went back up. Maybe a glitch or maybe I just got unlucky when I was attacking that somehow I in my button mashing of attacks she defended herself. Whatever, it's easy once you know.
Last bit just in case, once you start attacking and you know she's taking hits don't stop. Keep hitting X and you should be good.
After that, you go into an office and have to hit her 3 times. She's initially sat at a desk and does nothing; so, it's easy Only issue is all the boxes in your way.