fix: preserve subpaths in SSE endpoint URL construction

[Njuelle]

Motivation and Context

When MCP Server is deployed under a subpath URL (e.g., http://domain.com/subpath/sse), clients fail to connect because of the way the SSE transport constructs URLs. Specifically, the use of new URL() incorrectly discards the subpath, transforming requests intended for /messages/ into http://domain.com/messages/, rather than preserving the correct subpath (http://domain.com/subpath/messages/).

This fix ensures that when a server is deployed at a non-root path, clients can still connect successfully by preserving the subpath in all URL communications.

How Has This Been Tested?

• Added unit tests for both client and server components that verify subpath preservation
• Added tests for various URL structures (complex nested paths, paths with query parameters, etc.)
• Verified through integration tests with actual HTTP requests
• Manually tested with Inspector connecting to a server deployed at a subpath

Breaking Changes

None. This is a pure bug fix that maintains backward compatibility. Existing deployments at root paths continue to work as before.

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

The fix involves two main changes:

1. Server-side: Properly handling URL construction to preserve subpaths when generating endpoint URLs
2. Client-side: Properly resolving relative URLs against the original URL to maintain subpath context

The fix handles edge cases including empty paths, invalid URLs, and absolute URLs with proper error handling.

fixes #470

[fvdcx]

@Njuelle thanks for doing this, can you involve the author to review the code change?