how to avoid cancellations?

That level of exclusivity should require a nonrefundable deposit since your profile clearly states that you accept other dogs. I would call Rover and see if there is any way it can be done. You would set the amount (let's say, 50%) and the customer would have to agree to pay that amount regardless of what happens.

In the future, you may not want to accept such a potentially money-losing situation when it means that you have to turn down others. An alternative is to create an environment where the unfriendly dog could live and be apart from the other dogs you accept.

I would not book myself off until the client has done a proper reservation and let the client know that until you are guaranteed to be paid, you cannot risk loosing the revenue. When the client does a reservation, the money is frozen on their card and charged once the stay is over, depending on your cancellation settings, if the client cancels, you will at least get some of the amount.

Since I am very clear about having many dogs in my home for both daycare and boarding, if a client's dog cannot be with other dogs, therefor making it so I cannot take more dogs, I ask for twice the amount. I know it sounds like a lot but this is my job and like any other business, I need the revenue so it is understandable that the client has to compensate for my lost revenue. .