test(fastapi): Verify request info captured with POST endpoints

[alexander-alderman-webb]

Note: Depends on #6282

Description

Adapt JSON and FormData POST request body tests from the Starlette integration test suite.

Issues

Reminders

• Please add tests to validate your changes, and lint your code using tox -e linters.
• 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 📊

✅ 282 passed | Total: 282 | Pass Rate: 100% | Execution Time: 41.52s

📊 Comparison with Base Branch

Metric	Change
Total Tests	—
Passed Tests	—
Failed Tests	—
Skipped Tests	—

✨ No test changes detected

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 14800 uncovered lines.
✅ Project coverage is 33.6%. Comparing base (base) to head (head).

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    33.53%    33.60%    +0.07%
==========================================
  Files          190       190         —
  Lines        22342     22288       -54
  Branches      7576      7544       -32
==========================================
+ Hits          7492      7488        -4
- Misses       14850     14800       -50
- Partials       749       747        -2

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