test(fastapi): Verify request info capture 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 📊

✅ 88759 passed | ⏭️ 6019 skipped | Total: 94778 | Pass Rate: 93.65% | Execution Time: 299m 21s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📈 +39
Passed Tests	📈 +39
Failed Tests	—
Skipped Tests	—

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 2360 uncovered lines.
✅ Project coverage is 89.9%. Comparing base (base) to head (head).

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.90%    89.90%        —%
==========================================
  Files          192       192         —
  Lines        23369     23369         —
  Branches      8038      8038         —
==========================================
+ Hits         21009     21009         —
- Misses        2360      2360         —
- Partials      1329      1329         —

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