Cat follow up: Inappropriate Elimination?

Jesus… this new cat is causing quite a stir in my household. I am now using Feliway around my home, and it has helped ease some of the tensions between my two cats, but…

My new cat poops on the rug in the guest bathroom every night between 3–6 AM. Right on the middle of the rug. I took the rug away, and she pooped right where the rug WOULD be.

I’ve resorted to just trying to close the door, but I don’t want to be embarrassed by a guest finding an unwelcome surprise when they forget to close the door. Are there any behavior modification techniques that you have tried to stop inappropriate elimination? Any leads are welcome. Thanks, flutherers.

Addendum: My cat DOES use her litterbox for everything else. The problem is just this one early-morning BM.