Ehhhh....not a good idea, IMO......paper staples won't got thru corrugated far enough to stay in the wood for long........and they might not eat cardboard but you sure don't want them to eat staples.
Might work to cut it the juuust right size to jam in tight between top and bottom of nest.
Ahh I didn't think about that, that they could pull out the staples. With the top opening lid it'd be a pain to try and keep them in place every time you open the lid though. Maybe folding cardboard pieces in half, like a steep A shape, and sort of sitting them over the existing dividers. I'd either wrap them securely in wire close to where the top of the dividers currently end, to help keep them in place (so they sort of pinch down on the wood), or punch holes in the cardboard just above the current dividers and zip tie through the holes to get the same effect?