Update: renewed US passport from UK without knowing SSN, it worked
Quick follow up since my original post got removed. Short version of the original: my partner is American, lived in London about 10 years, passport expired last week, and she had no idea what her SSN was. Lost the card years ago, no US documents over here, nothing to dig through. We were stuck. Here is what actually worked.
She booked an in person appointment at the Federal Benefits Unit at the US Embassy in London. Getting the appointment was the hardest part honestly, contacting them is a nightmare, but once she had a slot the rest was straightforward.
She brought her expired US passport as ID. That was it. They verified her identity, looked her up in the system, and told her the SSN verbally at the appointment. Walked out with the number written down. From there the passport renewal by mail is a normal DS-82 since hers expired under 15 years ago.
If anyone is in the same boat, the FBU appointment is the answer. Do not bother trying to renew the passport with the SSN field blank or marked unknown, I called the passport centre and they confirmed they will reject the application. You need the actual number. The FBU can give it to you.