fix(docker): Pre-install Playwright Chromium browsers for automated testing

Fixes #725

AI agents in automated testing mode require Playwright to verify implementations,
but Docker containers had only system dependencies installed, not browser binaries.
This caused verification failures with permissions errors.

Changes:
- Install Playwright Chromium in Dockerfile (~300MB increase)
- Update docker-compose.override.yml.example with clearer Playwright documentation
- Add "Playwright for Automated Testing" section to README
- Document optional volume mount for persisting browsers across rebuilds

Browsers are now pre-installed and work out of the box for Docker users.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

README.md

@@ -338,6 +338,33 @@ services:
 
 The Docker image supports both AMD64 and ARM64 architectures. The GitHub CLI and Claude CLI are automatically downloaded for the correct architecture during build.
 
+##### Playwright for Automated Testing
+
+The Docker image includes **Playwright Chromium pre-installed** for AI agent verification tests. When agents implement features in automated testing mode, they use Playwright to verify the implementation works correctly.
+
+**No additional setup required** - Playwright verification works out of the box.
+
+**Optional: Persist browsers across container rebuilds**
+
+To avoid re-downloading browsers when rebuilding the Docker image, add this to your `docker-compose.override.yml`:
+
+```yaml
+services:
+  server:
+    volumes:
+      - playwright-cache:/home/automaker/.cache/ms-playwright
+
+volumes:
+  playwright-cache:
+    name: automaker-playwright-cache
+```
+
+**Updating browsers manually:**
+
+```bash
+docker exec automaker-server npx playwright install chromium
+```
+
 ### Testing
 
 #### End-to-End Tests (Playwright)