[FEATURE] VSCode Extension login is derived from base VSCode instance

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

When you login to vscode's claude code extension, it works, and then you login to a remote server and it's time to login again. Every time that you login to a dev container, it's time to log in again. This is especially annoying if you're not on an API key, because then you have to go through the login flow every time, website pop up, etc.

The VSCode extension should have a mechanism for login in once on the base VSCode instance, and then whenever you connect to a remote machine or dev container, the login information can optionally be forwarded. Otherwise the vscode extension is unusable in situations where you're remote-ing into lots of different machines.

Proposed Solution

I'd like to be able to log into the vscode extension once on the local app on my laptop, and then have this session be available when I remote onto different machines or run dev containers.

Alternative Solutions

Not use the vscode plugin at all, have claude ssh into the remote machines from my local terminal so it stays logged in. This is pretty bad and defeats the purpose of having a vscode extension.

Priority

High - Significant impact on productivity

Feature Category

Configuration and settings

Use Case Example

I have to remote into a bunch of remote machines to do dev work and I remote into the machine, setup a dev container and then have to login again. I'm only on the machine for a little while, so this has me logging in all the time. I really should only have to log in once, on the base machine, like I do for GitHub.

Additional Context

_No response_