Hi Colum, it seems to me this is a matter of the relative resolution setting for your second monitor. Try adjusting it to maybe 120%, or 150%, so the icons and other graphic features becomes smaller. I think the reason why the Animation Paper window can’t be resized smaller is because it is already at its minimum size. Also its UI must be quite large with your setup. Adjusting the size in your Windows setup will help with this issue as well.
Let me know how it goes?