ci: Remove pytest-watch from requirements-testing.txt

[alexander-alderman-webb]

Description

Remove the package from requirements-testing.txt and add the package as a dependency to only the test suites that rely on the package (if applicable).

This is part of environment cleanup before moving the remaining testing dependencies to a uv dependency group.

Issues

Reminders

• Please add tests to validate your changes, and lint your code using uv run ruff.
• Add GH Issue ID & Linear ID (if applicable)
• PR title should use conventional commit style (feat:, fix:, ref:, meta:)
• For external contributors: CONTRIBUTING.md, Sentry SDK development docs, Discord community

[github-actions]

Codecov Results 📊

✅ 91388 passed | ⏭️ 6136 skipped | Total: 97524 | Pass Rate: 93.71% | Execution Time: 333m 59s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📈 +29
Passed Tests	📈 +29
Failed Tests	—
Skipped Tests	—

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 2411 uncovered lines.
✅ Project coverage is 89.85%. Comparing base (base) to head (head).

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.85%    89.85%        —%
==========================================
  Files          192       192         —
  Lines        23758     23758         —
  Branches      8204      8204         —
==========================================
+ Hits         21346     21347        +1
- Misses        2412      2411        -1
- Partials      1345      1344        -1

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