Here is how I read the following:
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.
(8) Subsection (1)(c) of this section does not apply to any administrator or employee of the facility or to any person who, upon entering the place or facility, directly and promptly proceeds to the administrator of the facility or the administrator's designee and obtains written permission to possess the firearm while on the premises.
(10) Any person violating subsection (1) of this section is guilty of a gross misdemeanor.
You will know because the facility is required by subsection (5) to post signage if it is a banned facility. From your explanation and what I read above, it does not sound like you will be in a banned facility. As it must be a certified facility. However, please check and verify for yourself. Note, that a mental health facility as defined by 9.41.300 is not required to provide lock boxes or a designated official for safe keeping of a firearm. Only subsection (1b) requires this clause (i.e. courts). Also note that you may pursue written permission to carry. Perhaps a licensed armed guard in the state of Washington would be compelling for the administrator to grant permission.