Correct. Where the property is rented out, as it is in this case for LAHC, the owner is liable on behalf of the tenant. It is up to the owner to recover any costs from their own tenant under the RTA if they choose to. Basically, you only need to deal with the owner, not the tenant.
Something else to think about too, is that a wooden fence may not be appropriate due to the tenant's dog(s). The fence may need to be upgraded to colorbond in order to be "suitable" in this case, and may also need a full length concrete footing to stop the dog(s) from digging under it. Many LAHC homes already have colorbond fences anyway, so it's not unreasonable to ask.
Something else to think about too, is that a wooden fence may not be appropriate due to the tenant's dog(s). The fence may need to be upgraded to colorbond in order to be "suitable" in this case, and may also need a full length concrete footing to stop the dog(s) from digging under it. Many LAHC homes already have colorbond fences anyway, so it's not unreasonable to ask.