fix: browse env inconsistently updates internal state (browserbase#1806)

## Summary
- `browse env` showed stale "local" mode after `browse env remote`
- Root cause: `.mode` file was only written during lazy browser init
(`ensureBrowserInitialized`), not at daemon startup. Between daemon
start and first command, `readCurrentMode()` returned `null` and fell
back to hardcoded `"local"`
- Write `.mode` eagerly in `runDaemon()` at startup so it's immediately
available
- Fall back to `desiredMode` instead of `"local"` in the `env` display
handler as a safety net

## Test plan
- [x] Reproduced bug: `browse env remote` → `browse env` showed
`"mode":"local"`
- [x] Verified fix: `browse env remote` → `browse env` now shows
`"mode":"remote"`
- [x] `mode.test.ts` passes (3/3)


<!-- This is an auto-generated description by cubic. -->
---
## Summary by cubic
Fixes `browse env` showing stale "local" after `browse env remote`
(STG-1547). The daemon now writes `.mode` at startup, the display falls
back to `desiredMode` until mode is written, and a patch changeset is
added for `@browserbasehq/browse-cli`.

<sup>Written for commit 9661d92.
Summary will update on new commits. <a
href="https://cubic.dev/pr/browserbase/stagehand/pull/1806">Review in
cubic</a></sup>

<!-- End of auto-generated description by cubic. -->

---------

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: shrey150

.changeset/fix-cli-env-mode.md

@@ -0,0 +1,5 @@
+---
+"@browserbasehq/browse-cli": patch
+---
+
+Fix `browse env` showing stale mode after `browse env remote`

packages/cli/src/index.ts

@@ -260,8 +260,9 @@ const DEFAULT_VIEWPORT = { width: 1288, height: 711 };
 async function runDaemon(session: string, headless: boolean): Promise<void> {
   await cleanupStaleFiles(session);
 
-  // Write daemon PID file
+  // Write daemon PID file and initial mode so status is immediately available
   await fs.writeFile(getPidPath(session), String(process.pid));
+  await fs.writeFile(getModePath(session), await getDesiredMode(session));
 
   // Browser state (initialized lazily on first command)
   let stagehand: Stagehand | null = null;
@@ -1537,7 +1538,7 @@ program
       let mode: string | null = null;
       const desiredMode = await getDesiredMode(session);
       if (await isDaemonRunning(session)) {
-        mode = toModeTarget((await readCurrentMode(session)) ?? "local");
+        mode = toModeTarget((await readCurrentMode(session)) ?? desiredMode);
       }
       console.log(JSON.stringify({
         mode: mode ?? "not running",