Fix: Increase the default http timeout for wp/v2/settings for put and post request

[hbhalodia]

What?

Closes #940

Why?

• PR resolves the issue where settings save is removing the openAI key while checking the authentication on each save request, which sometimes throws the timeout error.

How?

• PR adds the support of increased http timeout on filter which only added when the route called is wp/v2/settings.

Use of AI Tools

• Yes, Claude Code, Opus 5.
• Used for timeout implemenation and for unit tests to add all scenarios.

Testing Instructions

1. Add the openAI key in connectors screen.
2. Go to Settings --> AI.
3. Quickly update some settings multiple times.
4. Check for response, it should not remove the openAI key.

Changelog Entry

Fixed - Prevent stored AI provider API keys from being cleared during unrelated /wp/v2/settings saves by extending the HTTP timeout while core revalidates them.

[github-actions]

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Co-authored-by: hbhalodia <hbhalodia@git.wordpress.org>
Co-authored-by: dkotter <dkotter@git.wordpress.org>

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 74.42%. Comparing base (788e1a6) to head (58839c5).

Additional details and impacted files

@@              Coverage Diff              @@
##             develop     #947      +/-   ##
=============================================
+ Coverage      74.39%   74.42%   +0.02%     
- Complexity      3110     3116       +6     
=============================================
  Files            132      132              
  Lines          12166    12179      +13     
=============================================
+ Hits            9051     9064      +13     
  Misses          3115     3115              

Flag	Coverage Δ
unit	74.42% <100.00%> (+0.02%)	⬆️

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