Send custom user agent header on Studio Code AI requests

[sejas]

How AI was used in this PR

AI was used to investigate where Studio Code's AI requests originate, implement the header change, and update the affected tests. The change was reviewed and verified locally.

Proposed Changes

• Use custom user agent instead of default one.

Testing Instructions

1. Run npm run cli:build && node apps/cli/dist/cli/main.mjs code and start a Studio Code chat on the WordPress.com provider.
2. Confirm you get a 403 error. I'll whitelist the new UA before merging this PR.

Pre-merge Checklist

• Have you checked for TypeScript, React or other console errors?

[wpmobilebot]

📊 Performance Test Results

Comparing 2ce4476 vs trunk

app-size

Metric	trunk	2ce4476	Diff	Change
App Size (Mac)	1402.84 MB	1402.84 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	2ce4476	Diff	Change
load	1064 ms	1057 ms	7 ms	⚪ 0.0%

site-startup

Metric	trunk	2ce4476	Diff	Change
siteCreation	7533 ms	7523 ms	10 ms	⚪ 0.0%
siteStartup	2865 ms	2874 ms	+9 ms	⚪ 0.0%

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Results are median values from multiple test runs.

Legend: 🟢 Improvement (faster) | 🔴 Regression (slower) | ⚪ No change (<50ms diff)