test(browse): run watchdog test servers under a private HOME

The headed-mode case starts a real server with BROWSE_HEADED=1, and its
headed launch writes ~/.gstack/.auth.json for the extension. With the
caller's HOME, a slow kill let that write reach the real home and failed the
free runner's home-containment check (free-suite shard 5 on #3075). Give each
spawned server a private HOME and GSTACK_HOME, keeping Playwright's browser
cache path.

Co-authored-by: capy-ai[bot] <230910855+capy-ai[bot]@users.noreply.github.com>

browse/test/watchdog.test.ts

@@ -51,9 +51,19 @@ afterEach(async () => {
 
 function spawnServer(env: Record<string, string>): Subprocess {
   const stateFile = path.join(tmpDir, 'browse-state.json');
+  const home = path.join(tmpDir, 'home');
+  fs.mkdirSync(home, { recursive: true });
   return spawn(['bun', 'run', SERVER_SCRIPT], {
     env: {
       ...process.env,
+      // The headed server writes ~/.gstack/.auth.json for its extension.
+      HOME: home,
+      USERPROFILE: home,
+      GSTACK_HOME: path.join(home, '.gstack'),
+      PLAYWRIGHT_BROWSERS_PATH: process.env.PLAYWRIGHT_BROWSERS_PATH || path.join(
+        process.env.XDG_CACHE_HOME || path.join(os.homedir(), process.platform === 'darwin' ? 'Library/Caches' : '.cache'),
+        'ms-playwright',
+      ),
       BROWSE_STATE_FILE: stateFile,
       BROWSE_PORT: '0', // Use the existing available-port allocator; fixed ports can collide across shards.
       ...env,