Fix flaky SSE tests: test_sse_client_basic_connection_mounted_app and test_request_context_isolation

[maxisbey]

Initial Checks

• I confirm that I'm using the latest version of MCP Python SDK
• I confirm that I searched for my issue in https://github.com/modelcontextprotocol/python-sdk/issues before opening this issue

Description

Two SSE tests in tests/shared/test_sse.py are flaky:

Affected tests:

• test_sse_client_basic_connection_mounted_app (line 326)
• test_request_context_isolation (line 462)

CI evidence: https://github.com/modelcontextprotocol/python-sdk/actions/runs/19330649603/job/55292600721?pr=1586

Test locations:

• tests/shared/test_sse.py:326 - test_sse_client_basic_connection_mounted_app
• tests/shared/test_sse.py:462 - test_request_context_isolation

Related Issues

• Bug Report: FastMCP RuntimeError: Received request before initialization was complete Leading to Empty SSE Responses When Embedded in FastAPI #737 - RuntimeError: Received request before initialization was complete (SSE)
• Fix flaky test test_response in test_streamable_http.py #1573 - Similar flaky test pattern in streamable HTTP (404 instead of 200)

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