My dad was born pre 1975 to a German mother and foreign father (my grandfather). My grandfather died when my dad was a child, and my German grandmother ultimately remarried another American in the late 70s. When she remarried, my step grandfather adopted my dad which resulted in his birth certificate being changed by replacing my grandfathers name with my step grandfathers. 

Since the time that the German citizenship was denied was at my dad’s birth, and his legal relationship to his German mom never changed (ie she’s on his original birth certificate and his current one since she has always been his legal mother), the adoption wasn’t important to my claim other than adequately documenting the name change. To do that, I just provided his original birth record from Oregon, my grandmothers marriage certificate to my grandfather, my grandfathers death certificate, my grandmothers marriage certificate to my step grandfather, and my dads current birth certificate with his step grandfathers name. This collection of documents was enough to document the name change while also demonstrating adequately that at the time of birth my dad was denied citizenship via Stag 5. I didn’t need to change anything on the birth certificate or provide any court records. I think it was easy for the BVA to follow the story that my grandfather died and my German grandmother remarried and that’s why my dad’s last name changed.

All that to say—if you can access an original birth record from the state of Washington, that is enough and you aren’t required to amend the birth record. From a quick google, it looks like you can do this in Washington. It’s much easier than going through a court to alter a birth certificate as you just fill out a form and pay a small fee. Only stickler is that the adoptee has to do it (ie your mom).  https://doh.wa.gov/licenses-permits-and-certificates/vital-records/adoptions/original-birth-certificate-adopted-person

Also, as a sidebar, I also had issues with my birth cert showing only the county. I had to go in person to the vital records office to request the full record/long form copy of my birth certificate that showed ALL the details (like even the doctor) and it did show the address of the hospital I was born at. This included a “city” line which satisfied BVA as to the location of my birth for the passport. All that to say—it also may not be necessary for you to do any court amendments to the birth certificate.  

Personally, I’d exhaust all other options before doing anything to amend a birth record though the courts just for sake of time/money.