UPDATED: smoother and thicker!

Insert a half-inch dowel into the caps, fit the threads through half-inch spaced cage bars, and fix it in place with the nuts. The nicest thing about this design is that you your dowel does not need to be cut perfectly.

It worked for me to decrease the size of the awesome "Knurling bolt and nut" by akira3dp0 - you could increase it for a larger dowel/larger cage. Does not require supports.

If you have a different-sized dowel you can edit this item yourself online for free in Tinkercad: https://www.tinkercad.com/things/0B1wd431lAN

Print in PET if you can for least toxicity in case your bird chews on the parts. Print two bolts and two nuts, one set for each end of the perch.