Restarting actions (update, experimental toggles) don't preserve session

### Describe the bug

When I start Copilot CLI, and perform an action that restarts the app (such as update or toggling experimental mode) it does not restart correctly, and gives an error message that the session ID cannot be found:

```
Error: No session, task, or name matched 'b490e4ae-7edc-4d79-bfb0-48a03c10084f'.

To resume by session or task ID: copilot --resume=
To resume by name: copilot --resume=
To name a new session: copilot --name=
To pick from existing sessions: copilot --resume
```

### Affected version

GitHub Copilot CLI 1.0.51-2

### Steps to reproduce the behavior

1. Type `copilot` to start the cli
2. `/experimental on` (or off if you already have it on)

### Expected behavior

Copilot CLI restarts correctly enabling experimental.

What actually happens is the error I listed above.

### Additional context

Logs from the relaunch:

```
2026-05-20T18:32:29.847Z [DEBUG] Sending telemetry event: copilot-cli/extension.activate
2026-05-20T18:32:29.848Z [INFO] Shell completions: no supported shell detected, skipping
2026-05-20T18:32:29.849Z [DEBUG] OpenTelemetry not enabled (resolveOtelConfig returned null)
2026-05-20T18:32:29.881Z [DEBUG] No persisted remote state for session 796be1bf-039a-4cfa-89fb-6dc690047291
2026-05-20T18:32:30.090Z [INFO] Session indexing debug: SESSION_INDEXING=true, repository=333fred/roslyn
2026-05-20T18:32:30.304Z [DEBUG] ExP request: GET https://telemetry.enterprise.githubcopilot.com/telemetry?tas-endpoint=githubdevdiv
2026-05-20T18:32:30.304Z [DEBUG] ExP headers: {"User-Agent":"copilot-cli","X-ExP-SDK-Version":"1","X-VSCode-ExtensionName":"CopilotCLI","X-MSEdge-ClientId":"","x-exp-parameters":"copilottrackingid=,github_copilotcli_cliversion=1.0.51.2,github_copilotcli_prerelease=1,github_copilotcli_audience=microsoft,github_copilotcli_experimentationoptin=0,extensionname=CopilotCLI,github_copilotcli_firstlaunchat=1773187200,github_copilotcli_copilotplan=enterprise"}
2026-05-20T18:32:30.569Z [DEBUG] Sending telemetry event: copilot-cli/cli.telemetry (kind: exp_context_fetch)
2026-05-20T18:32:30.721Z [ERROR] Request to Copilot Task API at https://api.enterprise.githubcopilot.com/agents/tasks/796be1bf-039a-4cfa-89fb-6dc690047291 failed with status 404 (request ID: A8EA:2636CF:65A2510:6D113F7:6A0DFE3E), body: {"documentation_url":"https://docs.github.com/rest","message":"not found"}

2026-05-20T18:32:31.351Z [INFO] No update needed, current version is 1.0.51-2, fetched latest release is v1.0.51-2
2026-05-20T18:32:32.942Z [DEBUG] Fetched 80 remote sessions
```

Beginne damit, die Aktion `/experimental` und den von ihr aufgerufenen Neustartpfad nachzuverfolgen, und folge dann der Übergabe der aktuellen Sitzungs-ID an den erneuten Start. Reproduziere das Problem mit `/experimental on` oder `off` und untersuche die Protokolle des erneuten Starts. Erledigt ist die Aufgabe, wenn die CLI mit der angeforderten Einstellung neu startet und die aktive Sitzung ohne einen Fehler wegen einer fehlenden Sitzung beibehält.

Vom Indexierungsmodell aus dem Issue-Text verfasst.