It is kind of nice that this is the policy, if indeed it is. Keeps us from declining people over and over who won't take no for an answer.
You might want to just tell them sorry, nothing you can do, the system does not allow, etc. They, and you, will be happier if they can find a pet-friendly host.
If they are the sort who will keep trying anything, not at all concerned that you will get very ill, they may just be telling you what you want to hear, and low and behold will show up with the dogs? Just a thought (based on some experience...)