HMMMM - made me think of something, might sound CRAZY...but how about putting a bit of barbed wire around the bottom of the bird cage?

Or something pointy that won't really hurt, but will discourage - like the hedgie quill!

Also - cats hate the smell of vinegar, so damping a rag with it and putting it there might help...if the birds are ok with it, that is!

Good luck!

Catty1