I went down this rabbit hole a bit recently trying to use claude inside fence[0] and it seems that on macOS, claude stores this token inside Keychain. I'm not sure there's a way to expose that to a container... my guess would be no, especially since it seems the container is Linux, and also because keeping the Keychain out of reach of containers seems like it would be paramount. But someone might know better!
0: https://github.com/Use-Tusk/fence