Should I change rates in the middle of a stay?

I agree that it's best not to attempt any refunds/adjustments until the stay is complete. If I were in your position, I would hesitate to offer any kind of refund, and I probably wouldn't refund the entire difference between cat care only and cat + dog care if the client were to ask. On the one hand, I understand feeling weird about getting paid for services you aren't providing. On the other, this arrangement wasn't made ahead of time when you had the opportunity to agree to these terms. On any given day you don't know what services you'll be providing when you arrive. You're still traveling the full distance each way, and whether or not the dog is there, you're budgeting your time to allow for dog care. You're still planning your day around taking care of the dog, but looking at potentially only being compensated for taking care of the cat.

Since you have a flexible cancellation policy for drop-in visits, a customer can get a full refund if the service is canceled anytime before it is to be provided. There is no need to adjust any rates since the entire engagement has been (or should have been) paid for upfront. You are going there anyway to take care of the cat. At the end of the engagement, you can issue a refund for when the dog was not present. If the customer asks about it, tell her that you'll put through an adjustment at the end.

I think part of the consideration is your time, and if you counted on dog sitting for this pet at this time, you don't need to refund the difference on such short notice. If you set aside the time to watch this dog, it can take away from time could could have spent sitting for another dog and earning that money since it sounds like it's too short of notice to find another client for those days.