Washington State Preemption covers this situation, per Washington Code 9.41.300.
Open Carry of a Loaded Handgun on City Property/Public Buildings is Legal throughout Washington, except for the Buildings with Restrictions mentioned under the Code cited above.
Under Washington State Law 9.41.300, the Buildings with Restrictions that are permissable under State Law are: 1. Jails/Law Enforcement Facilites/Place of Detention, 2. CourtHouses (EXCEPT areas of engress and igress, per a LockBox), 3. Public Mental Health Facility (areas of engress and igress are exempt again), 4. Areas off-limits under for Persons under 21 years of age in Licensed Liquor Establishments (NOT an Establishment that merely serves Alcohol, but ONLY the Portion devoted to the age Restriction so cited), 5. Restricted TSA Screening Checkpoint in International Federally Protected Airports for Safe Passage in International Travel (NOT Merely Just The Terminals), and 6. Local Convention Centers or Stadiums (EXCEPT for Persons with Concealed Pistol License).
Persons who are Licensed pursuant to a Valid Washington Concealed Pistol License may Check-in their Weapons upon arrival at any place enumerated in 1, BUT they [The Weapons] must surrender their Weapons upon arrival and depart immediately therefrom when they [The Weapons] are returned upon departure.