Fix: Fix MCP_PROXY_FULL_ADDRESS base path handling for SSE message endpoint

[ronantakizawa]

Summary

Fixes inconsistent handling of MCP_PROXY_FULL_ADDRESS base paths between SSE and message endpoints.
When using a proxy with a base path (e.g., /inspector/mcp_proxy), the SSE endpoint correctly respects
the full address while the message endpoint incorrectly strips the base path.

Motivation and Context

This change fixes issue #630 where the MCP_PROXY_FULL_ADDRESS environment variable is inconsistently handled between SSE and message endpoints, breaking SSE transport functionality when deployed behind reverse proxies with base paths. When MCP_PROXY_FULL_ADDRESS is set to a URL with a base path (e.g., https://example.com/inspector/mcp_proxy), the SSE transport becomes completely non-functional due to inconsistent URL construction. This results in SSE connections establishing successfully, but all subsequent message requests failing
with 404 errors because they're sent to the wrong URL.

Solution:
Extract the base path from MCP_PROXY_FULL_ADDRESS and apply it consistently to both SSE and message
endpoints, ensuring SSE transport works correctly with reverse proxy deployments.

Before (Broken):

• ✅ SSE requests: https://example.com/inspector/mcp_proxy/sse (correct)
• ❌ Message requests: https://example.com/message (incorrect - base path stripped)

After (Fixed):

• ✅ SSE requests: https://example.com/inspector/mcp_proxy/sse
• ✅ Message requests: https://example.com/inspector/mcp_proxy/message

How Has This Been Tested?

• Unit Tests: All existing tests pass (289/289 client tests)
• Build Verification: Full build pipeline successful
• Logic Verification: Tested path extraction logic with various URL formats:
  • No env var → /message
  • http://localhost:6277 → /message
  • http://localhost:6277/inspector/mcp_proxy → /inspector/mcp_proxy/message
  • http://localhost:6277/inspector/mcp_proxy/ → /inspector/mcp_proxy/message
• Integration Testing: Verified server starts correctly with base path configuration

Breaking Changes

None. This is a backward-compatible bug fix:

• Existing deployments without MCP_PROXY_FULL_ADDRESS continue working unchanged
• Only affects SSE message endpoint path construction when base path is configured

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

Additional context