As I stated, call the builder. Using a Nerf blade (if screws can't be removed) would give you six pieces. That being the top, sides, back, front and bottom. Then I would screw an glue it back together. If it was steel an welded, the welds would be cut an reweld. If it was built it can be taken apart. Heck, today molded stuff can be cut up an put back together without showing a seam. There are a ton of ways to "fix" your problem. Some would require more skill then others. You could turn the cage into a modular set up. A saw, some channel, screws an caulk would give you a three piece cage.
Removing a door an frame (an jack studs if needed) could be easier. Talk to the builder (first) an any carpenters (framers if removing a door, cabinet guys for the cage) you know. Work smart not hard!
Good luck!