How much liability insurance coverage should a renter get for his large dog in Washington State?

Leo, 

Per the Dangerous Dog Law; yes the owner would be required to provide proof of insurance for their dog in the amount of $250k in order to license the K9.  Has the dog in your scenario been identified by the local authorizes as a "dangerous dog?"  If the dog is a "dangerous dog" and has attacked someone or something, more than likely the dog will make the renter uninsurable in the standard markets.  A surplus lines policy would have to be obtained.  If you are in the position of landlord you need to have a copy of that policy in your hand prior to renting the unit, and on a "going forward basis" you should establish your business practice when it comes to pets in general.

If there isn't a dangerous dog in the equation, we advise our clients to carry at least $500,000 Liability Limits.  It does not matter if they are a homeowner or a renter.  Your liability exposure it high either way.  Remember the liability extends to the insured's animals (as long as they are approved by the insurance company - NO AGGRESSIVE BREEDS), acts of insured's children up to the age of 13, sports the insured plays and volunteer work to name a few.  Another aspect to remember is the Liability Coverage is the cheapest coverage per $1000 in the whole policy. 

It is in everyone's best interest to have at least $500,000 Liability coverage, and if you are a landlord, you should mandate your renters to carry insurance.  Average Renters Policy in Washington State is about $150 a year, and many times less than that.

You are citing a statute for "dangerous dogs" that doesn't apply to most dogs.   Here is the Washington state definition of "dangerous dog".    Therefore,  renter's insurance and renter's liability insurance are reasonable requests, but insurance limits equivalent to that required for a dangerous dog is not reasonable IMHO.