ref(mcp): Replace instance patches with middleware

[alexander-alderman-webb]

Description

Add a middleware instead of wrapping handlers on registration.
All requests go through the middleware, whereas there are multiple ways to register a handler.

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 📊

✅ 90302 passed | ⏭️ 6310 skipped | Total: 96612 | Pass Rate: 93.47% | Execution Time: 313m 20s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📈 +265
Passed Tests	📈 +257
Failed Tests	—
Skipped Tests	📈 +8

All tests are passing successfully.

✅ Patch coverage is 83.55%. Project has 2429 uncovered lines.
✅ Project coverage is 89.86%. Comparing base (base) to head (head).

Files with missing lines (1)

File	Patch %	Lines
sentry_sdk/integrations/mcp.py	83.55%	⚠️ 25 Missing and 25 partials

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.74%    89.86%    +0.12%
==========================================
  Files          192       192         —
  Lines        23823     23951      +128
  Branches      8226      8284       +58
==========================================
+ Hits         21378     21522      +144
- Misses        2445      2429       -16
- Partials      1349      1368       +19

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[cursor]

Cursor Bugbot has reviewed your changes and found 1 potential issue.

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