You are legally allowed to keep your daughter's therapy dog in your rental house in San Francisco, even though your lease has a "no pets" policy. The Fair Housing Act (FHA) and the California Fair Employment and Housing Act (FEHA) both prohibit landlords from discriminating against tenants with disabilities, and this includes refusing to allow them to have emotional support animals.
To qualify as an emotional support animal, the dog must be prescribed by a mental health professional to treat a disability. In your case, your daughter's psychiatrist has written a letter verifying that the dog is a therapy dog. This is sufficient documentation to show that the dog is a reasonable accommodation for your daughter's disability.
Your landlord cannot force you to move out when your lease is up simply because you have a therapy dog. However, the landlord may be able to evict you if the dog causes a nuisance or damage to the property. If this happens, you will need to prove that the dog is not a nuisance or a danger.