E2E Tests: Fix post types setup plugin

[ockham]

While working on #397, I needed to change the add_new label on the custom post type (CPT) used for tests. However, this change didn't seem to take effect on the frontend.

This was due to the way the test CPT was registered: It was persisted to the DB, with another DB option set as a flag so the test wouldn't attempt to recreate the CPT and instead use the persisted one. As a result, any changes made programmatically to the CPT didn't take effect.

This PR changes the test setup so the CPT is created on the fly by the plugin (and not persisted to the DB).

Use of AI Tools

Claude Code was used to investigate the issue, and to write the fix.
(This is only fair: The issue was originally also produced by Claude Code.)

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 53.21%. Comparing base (2875e2c) to head (c09ea34).

Additional details and impacted files

@@             Coverage Diff              @@
##              trunk     #401      +/-   ##
============================================
- Coverage     53.80%   53.21%   -0.59%     
  Complexity     4423     4423              
============================================
  Files           298      298              
  Lines         39468    39468              
============================================
- Hits          21235    21003     -232     
- Misses        18233    18465     +232     

Flag	Coverage Δ
e2e-js	45.84% <ø> (-0.02%)	⬇️
e2e-php	41.20% <ø> (-1.24%)	⬇️
javascript	15.07% <ø> (ø)
phpunit	29.95% <ø> (ø)

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[github-actions]

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