fix: propagate contextvars to MCP client background thread

[cagataycali]

Description

This PR fixes the issue where MCPClient does not propagate contextvars to its background thread, causing context variables set in the calling thread to be inaccessible in custom transports or auth logic.

Related Issues

• Fixes [BUG] MCP Client does not propagate contextvars #1440
• Related: [BUG] Tools operating in different context and breaking AgentCore 3LO #946, [BUG] Configurable Threads #769 (similar issues fixed for ThreadPoolExecutor in PR share thread context #1146)

Root Cause

In MCPClient.start(), the background thread is created without copying the current context:

# Before (problematic)
self._background_thread = threading.Thread(target=self._background_task, args=[], daemon=True)

This means any contextvars set in the calling thread are not accessible in the background thread.

Solution

Apply the same pattern used in PR #1146 (src/strands/_async.py) - copy the context and run the background task via context.run():

# After (fixed)
ctx = contextvars.copy_context()
self._background_thread = threading.Thread(target=ctx.run, args=(self._background_task,), daemon=True)

Documentation PR

No documentation changes required - this is an internal implementation fix.

Type of Change

Bug fix (non-breaking change which fixes an issue)

Testing

• I ran hatch run prepare
• The fix follows the exact same pattern that was tested and validated in PR share thread context #1146
• Tested scenarios:
  • Context variables set before MCPClient.start() are now accessible in the transport callable
  • Context variables are available for auth logic that runs in the background thread

Checklist

• I have read the CONTRIBUTING document
• I have added any necessary tests that prove my fix is effective or my feature works
• I have updated the documentation accordingly
• I have added an appropriate example to the documentation to outline the feature
• My changes generate no new warnings
• Any dependent changes have been merged and published

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PR created by strands-coder autonomous agent 🦆

[codecov]

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