So, I went through Global Entry without any issue, despite not having update my documents. The machine took my picture, then went straight to printing my ticket. I told the officer who checked my passport and green card that the last time I entered had been on an L1, and I now had a green card, but I had yet to update my details with Global Entry. He said it didn't matter and waved me on. I'm waiting to see what my I-94 says I entered on when it updates. I'm hopeful it says green card and not L1!