Finding an apartment in Korea which is fine with me having a cat is being insanely difficult and stressful...

Conversation basically goes like this:
- I really like this apartment. Is it ok if I have a cat?
> No, no pets allowed of any kind.
- Why not?
> Because they will damage the property and create a mess.
- What if I agree to pay for any repairs for damage they cause and have it professionally cleaned?
> Still no.
- Why not?
> Because they will damage the property and create a mess.

Repeat forever.