Thanks for the compliments everyone!
Tsanford is correct that I used plastic shower rod/closet rod brackets (just like these). Because I didn't have any screws short enough, I just used a healthy dose of hot glue to affix them to the sides of the cage. I know just using hot glue sounds crazy, but they stuck surprisingly well! The bottom (long) pole just rests on the brackets and is removable. It's a very snug fit with no gaps or cracks. The top pole is both resting and glued onto the brackets.
We weight-tested both poles to 12lbs without sagging, breaking, or popping out of their holders. Though personally I hope Gloria never gets THAT big.
I can provide some additional detail pictures if that would help.