feat: Add switch organization to project switcher

[charlesvien]

Problem

After #2364 the renderer knows every org the user belongs to, but there is no in-app way to switch between them. Users would have to log out or change orgs in the PostHog web app.

Changes

1. Add a "Switch organization" submenu to the bottom-left ProjectSwitcher dropdown
2. List every org from orgProjectsMap, mark the current one, hide the submenu when there is only one org
3. Fire useSwitchOrgMutation on click and navigate to task input on success
4. Short-circuit when a switch is already pending or the user clicks the active org

How did you test this?

manually

Publish to changelog?

no

[charlesvien]

Warning

This pull request is not mergeable via GitHub because a downstack PR is open. Once all requirements are satisfied, merge this PR as a stack on Graphite.
Learn more

• feat: Approve ai data processing in app for admins #2367
• feat: Add switch to org button on plan usage screen #2366
• feat: Add switch organization to project switcher #2365 👈 (View in Graphite)
• feat: Scope oauth tokens by org #2364
• main

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[greptile-apps]

T-Rex Logs

What T-Rex did

• Ran the existing ProjectSwitcher organization switcher tests after the UI change; all five tests passed.
• Added focused component-level checks for no-current-project rendering, rapid org selection, success-callback navigation ordering, and post-switch project fallback behavior.
• Observed a wrong-org fallback in the generated checks: with currentOrgId='org-2', a stale current project, and projects from both orgs still present, the project hook selected project '1' from org-1.
• Noted that the happy-path manual flow still works, but the stale project scenario could leave the app in the wrong organization.

Artifacts

Verbose vitest output for generated org switcher checks

• Provides the full test run results and timing details for the generated org switcher checks.

Generated targeted org switcher tests

• Holds the new targeted tests exercising no-current-project rendering, rapid org selection, and post-switch navigation paths.

_{Ran code and verified through T-Rex}

Prompt To Fix All With AI

Fix the following 1 code review issue. Work through them one at a time, proposing concise fixes.

---

### Issue 1 of 1
apps/code/src/renderer/features/sidebar/components/ProjectSwitcher.tsx:146-149
**Prevent wrong-org fallback**

When this switches orgs and navigates straight back to task input, `useProjects()` can run its fallback project selection while auth state is still settling. That hook flattens `orgProjectsMap` across every organization, so if the current project is stale or missing after switching to `org-2`, it can call `selectProject` for the first project from `org-1`. The main auth service then moves `currentOrgId` back to the org that owns that project, leaving the user in the wrong organization. Please constrain the fallback to the selected org, or wait/gate navigation until the switched org's project state is settled.

_{Reviews (1): Last reviewed commit: "navigate after org switch and guard doub..." | Re-trigger Greptile}