Harden browser Playground request transport

[chubes4]

Summary

• Prefer the browser Playground client-level request() transport before falling back to requestHandler.request().
• Preserve structured per-attempt diagnostics when Playground method transports fail.
• Extend the browser runtime smoke test to cover dual-transport priority, handler fallback, and failed request diagnostics.

Fixes #580.

Testing

• node --check packages/wordpress-plugin/assets/browser-runtime.js
• npm run browser-runtime-operation-smoke
• php tests/smoke-wordpress-plugin.php
• Studio Web live eval against this WP Codebox worktree with --plugin-source wp-codebox=/Users/chubes/Developer/wp-codebox@fix-browser-request-failure-diagnostics/packages/wordpress-plugin wrote /var/folders/lr/c_cmmt7s0592m4njz99v5yb40000gn/T/opencode/wp-codebox-580-live-eval/evidence.json. It no longer showed playground_request_failed; the run failed earlier at codebox_session_ready because Studio Web reported the prepared preview target was not browser-ready.

AI assistance

• AI assistance: Yes
• Tool(s): OpenCode (openai/gpt-5.5)
• Used for: Implemented the browser runtime request transport hardening, adjusted smoke coverage, ran verification, and drafted this PR description for Chris to review.