Hi,

Often, once a cat has a taste of the outdoors, it's very difficult to keep them in.

I can only think of 2 options in your case......locking the dog door and having to let the dogs in and out or having some type of outdoor enclosure for the cat where she can enjoy being outdoors safely.

Hope something will work out so you don't have to rehome your kitty.

Regards,
PHKitkat