1. Introduce another box that does the exact same thing as the current box does but stays and scrolls through like spectator mode has. Granted spectator mode's box isn't scrollable, this would be a big feature to add but understandable if not added because of it being a UI. A larger box works just as well.
2. Have a second box that shows stat changes, weather conditions, hazards, and finally terrain. This information is readily available to you in both console games and showdown. It is a key function that greatly allows you to keep track of the battle.
3. Move both your and your opponents Pokemon box to either side of the main box. Maybe even labelling who's is who
Here is my rough overview of what the new boxes could possibly look likewhen using a 1920x1080 monitor window maximised.