Unfortunately we encounter these type of people in many places. It is not beneficial to engage them. Anything you say will be interpreted in a way that exacerbates (makes worse) her hysterics.

Clearly you can not control your cats nature. There is no reason you should have to try to alter the behavior of yourself, your pets or your environment to please her. Even if you tried, as you have noticed it would make little difference.

My advice is to avoid this woman. If you do engage her there is nothing wrong with telling her you do not have time to talk to her or any other number of assertive, conversation ending phrases.