I can't speak towards using a bird cage, but I understand your question. I had my jax in a 24x24x48 cage, and he was using every inch of space. I doubled his cage size, and he still uses the entire cage but doesn't get so pouty about not getting free range time when I'm traveling for work.
I looked into bird cages and aviaries and never found any that really gave me more cubic footage than a screen cage for a reasonable price.
You could do a few things: get a second screen cage and attach them together to double his space (or set up a free range area and give him regular free range time but be prepared for the begging to come out, haha). You can also use secondary climbing options to reduce the amount of screen climbing. A combination of construction fabric, bird ladders, vines mounted vertically, and other accessories will help. I am constantly shifting things to fill in his preferred routes of travel. Yes, he still screen climbs, especially when he's trying to get my attention, but I haven't had any broken nails or screen rubs yet.
It's partly enrichment to prevent boredom, partly because its fun to watch him explore the small changes in his enclosure. It would be something that's cheaper to try before investing in a larger enclosure. That being said, i firmly believe that larger is better.
