Hi Paul,
Somewhere in your contract with the client, there should be a clause stating that your equipment should be secured, and not used by the client's staff, also that they should insure the equipment at their own expense to cover any damage caused by them using it, (this would cover the damage done to your kit by the dog, not sure if the dog would count as staff though
).
We also have 2 clients who take their dogs into the workplace, we just charge a premium for these sites, explaining to the client that it takes us longer to vacuum, removing dog hairs, and more frequent bag and filter changes.
Just bill them for the repairs as GaryJ suggested.
Regards,
Rob