feat(streaming): Make set_conversation_id API work

[sentrivana]

Description

In streaming mode, the set_conversation_id API doesn't currently do anything. Change that.

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 📊

✅ 95450 passed | ⏭️ 6340 skipped | Total: 101790 | Pass Rate: 93.77% | Execution Time: 336m 24s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📈 +135
Passed Tests	📈 +135
Failed Tests	—
Skipped Tests	—

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 2510 uncovered lines.
✅ Project coverage is 89.68%. Comparing base (base) to head (head).

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.65%    89.68%    +0.03%
==========================================
  Files          193       193         —
  Lines        24324     24331        +7
  Branches      8572      8576        +4
==========================================
+ Hits         21807     21821       +14
- Misses        2517      2510        -7
- Partials      1395      1395         —

---

Generated by Codecov Action