It sounds like your property is not the right fit for this family - you might want to tell them that. If you have a NO animal policy, then NO means NONE.
I don't know what you have written as your pet policy, but below are some excerpts from a Rental Agreement for an animal-inclusive property that address some animal things. Perhaps you or someone can glean something useful from them. NOTE: Animal, NOT pet!
2. USE AND OCCUPANCY.
The premises may be used for residential purposes only. Businesses, daycare, customer visits, manufacturing, food preparation/sale, laundry service, mechanical work, auto body work, pet-sitting, animal breeding, animal boarding, kennels, farm animals, etc. are not allowed. Computer-based work by Tenant or authorized occupant that does not cause increased visitors or traffic at the premises is acceptable.
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_____(initial) The premises will be occupied ONLY by Tenant and other authorized occupants/animals, listed below. Visiting or shared-custody children, stepchildren, and other anticipated frequent overnight occupants must be listed on the application, and listed on this Agreement as authorized occupants. The maximum number of occupants in this rental unit is _4_ humans and _2_ animals.
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6. RENTERS’ INSURANCE.
Landlord's insurance covers the structure only. It does not cover damage or loss of Tenant's personal property such as furniture, appliances, computers, TVs, clothing, food, or anything else.
_____ (initial) Tenant is required to carry Renter's Insurance which shall include a minimum of $100,000 liability coverage that includes animal's behaviors, as applicable. Failure to carry Renter’s Insurance will not place the burden on Management. Management is not responsible for Residents' or guests' property or vehicles. Management is only responsible for the repair of the structure and mechanicals and is not responsible for damages resulting to Tenant's possessions from broken or defective appliances, mechanicals, plumbing, structure, fire, flood or any other peril.
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10. ANIMALS.
NO dog, cat, bird, fish, reptile, or any other animal of any kind, permanent or visiting, indoor or outdoor, shall be permitted on the premises without prior written consent of Landlord.
_____ (initial) Any animal desired as an occupant requires the submission of an Animal Application and approval of the same by Landlord, before being allowed on the premises. Animal occupants are limited to dogs, cats, or fish in a maximum of 5-gallon size tank. (One fish tank counts as 1 animal, no matter how many fish are in it.) Any animal with a history of aggressive behavior will be prohibited. In no instance will more than TWO animals be allowed. Every animal (except fish) must be spayed or neutered, be at least one year old, and be kept current on all required vaccinations and registration.
Any approved animal(s) must be listed in this Agreement as an authorized occupant. Only the animal(s) listed will be permitted on the premises.
There ____Is / ____ is not an approved animal occupant at the time of the signing of this Agreement.
Describe: ___________________________________________________________________________________
Tenant will be fully liable for damages and/or injuries to property and/or people that may be caused by animal(s). Tenant's Renter's Insurance must contain coverage for the same. Tenant agrees that if their animal(s) should at any time become destructive, annoying, bothersome, or in any way a nuisance or danger to other animals or people, or the premises, Tenant will - upon notice of Landlord or law enforcement agency - immediately remove said animal(s) from the premises.
Any unapproved animal(s) found on the premises will be considered a breach of this Agreement, may be removed by Landlord or animal control and all the Animal Fees listed below, retroactive to the date of the occupancy of this Agreement, shall be paid immediately to Landlord. This breach may also lead to termination of tenancy.
If Tenant wants to add an animal after taking possession, or replace one that has died or left, they must submit a new Animal Application. Landlord must approve the new animal before it is listed as an authorized occupant and allowed on the premises.
If the premises is not an "Animal Inclusive Rental", the following Animal Fees apply:
If an animal is approved as an occupant, an additional security deposit of $250.00 (Total, for 1 or 2 animals. This amount is not included in clause 5 above), will be due to Landlord before the animal is permitted on the premises. Also, additional rent of $40.00 per animal per month will be paid at the same time, in the same manner, and with the same terms as all other rent.
This ____Is / ____ is not an "Animal Inclusive Rental".
--68.115.xxx.xx