fix: Read request body after handler so receive stream is not deprived

[alexander-alderman-webb]

Description

Ensure endpoint handler runs before the FastAPI or Starlette integrations read the request body.

As the integrations previously called high-level Starlette Request methods early, the underlying ASGI receive stream was consumed before the user's handler or middleware ran. While the Request object caches the request body, any handler or middleware that relies on low-level stream access still failed.

Issues

Closes #4764
Closes #4827
Closes PY-1851
Closes PY-1831

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

[linear]

PY-1851 Sentry FastAPI middleware hangs if user middleware raises an exception

PY-1831 Requests to fastapi-mcp hangs if Sentry is enabled

[codecov]

Codecov Report

❌ Patch coverage is 95.23810% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 84.83%. Comparing base (7ae6866) to head (f66bcd6).
⚠️ Report is 31 commits behind head on master.
✅ All tests successful. No failed tests found.

Files with missing lines	Patch %	Lines
sentry_sdk/integrations/starlette.py	90.62%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4832      +/-   ##
==========================================
+ Coverage   84.58%   84.83%   +0.25%     
==========================================
  Files         158      158              
  Lines       16521    16595      +74     
  Branches     2867     2877      +10     
==========================================
+ Hits        13974    14079     +105     
+ Misses       1703     1670      -33     
- Partials      844      846       +2     

Files with missing lines	Coverage Δ
sentry_sdk/integrations/fastapi.py	89.41% <100.00%> (+0.38%)	⬆️
sentry_sdk/integrations/graphene.py	90.12% <100.00%> (+0.37%)	⬆️
sentry_sdk/integrations/starlette.py	85.37% <90.62%> (+0.35%)	⬆️

... and 39 files with indirect coverage changes

[alexander-alderman-webb]

May need to change because the value associated with event["request"]["data"] was previously subject to request body size limits.

[alexander-alderman-webb]

Closing for the reasons in #4764 (comment)

[alexander-alderman-webb]

Reopening because there may be the option of changing data fields on transactions in a minor version.

Users with our FastAPI integration are facing empty receive streams when using fastapi_mcp or a custom middleware.