The Seattle Academy isn't on the list and literally held the state conference for GLSEN this year ( https://sites.google.com/glsenwa.org/2019-conference/home ).
> Catholic run schools (which are the majority of the private schools in the region)
Going to various directories that let you filter by religious affiliations shows that catholic schools are probably ~20% of the private schools in Kings County and ~15% across Washington State. Christian schools make up another ~25% in Seattle and ~20% statewide. So, while a significant portion, Catholic is definitely not the majority and the combination of Catholic+Christian might approach 50%.
I went to the first two schools alphabetically in NWAIS's directory of members for Seattle and they both had nondiscrimination policies that included sexual orientation and gender identity. Neither is Montessori or language immersion.
Bertschi (admittedly on the GBSA list also): https://www.bertschi.org/jobs
Northwest School
"Diversity is our commitment to build [a] community that includes individuals and families who represent a wide variety of ... gender identity and expression, sexual orientation"
Seattle Academy
"includes and embraces a spectrum of differences in ... gender, sexual orientation"
Lakeside
"Lakeside is committed to creating an inclusive and equitable community in which all individuals can participate in and contribute to the life of the school, regardless of ... gender, ... sexual orientation, or any other aspect of their identity."
My neighbor was fired over it and had trouble finding work at other private schools as there are just a handful of Montessori and language immersion schools, with everything else that isn't SPS having a Morals clause.
Edit: The GSBA directory only lists 2 schools, most private schools in Seattle are not supportive: http://thegsba.org/business-resources/guide-directory/single...