Show reasoning level in the model footer

[witbrock]

Summary

The model footer now shows the configured reasoning level and opens the existing language-model settings. It follows the same browser preference used for requests, falls back to the server-resolved preference, and shows “provider default” when no override is set. The label remains visible in the compact footer. Existing Settings persistence and provider capability controls are reused.

Validation

• 48 targeted tests pass (footer execution/identity, settings changes, local preferences).
• Frontend static lint and diff checks pass.
• Playwright using the real Settings template and modules with isolated fixture APIs: change low → high, reload, read persisted control/footer, inspect requested model parameters, clear to provider default. Label visibility checked at 1280px and 390px.
• Browser receipts/screenshots retained in .run/footer-reasoning/ in the task worktree. No live provider, database, or authentication acceptance is claimed.

Native task: #V#task_agent_7d69ab598ac4c4c83bc0ab6013651da8.
Merge decision: ready for this bounded UI claim. Deployment is not requested.