You picked a good one to look at.
Which one trumps the other?
RCW 72.23.300 Bringing narcotics, intoxicating liquors, weapons, etc., into institution or its grounds prohibited — Penalty.
Any person not authorized by law so to do, who brings into any state institution for the care and treatment of mental illness or within the grounds thereof, any opium, morphine, cocaine or other narcotic, or any intoxicating liquor of any kind whatever, except for medicinal or mechanical purposes, or any firearms, weapons, or explosives of any kind is guilty of a class B felony punishable according to chapter 9A.20 RCW.
RCW 9.41.300 Weapons prohibited in certain places — Local laws and ordinances — Exceptions — Penalty.
(1) It is unlawful for any person to enter the following places when he or she knowingly possesses or knowingly has under his or her control a weapon:
(c) The restricted access areas of a public mental health facility certified by the department of social and health services for inpatient hospital care and state institutions for the care of the mentally ill, excluding those facilities solely for evaluation and treatment. Restricted access areas do not include common areas of egress and ingress open to the general public;
(5) The perimeter of the premises of any specific location covered by subsection (1) of this section shall be posted at reasonable intervals to alert the public as to the existence of any law restricting the possession of firearms on the premises.