fix: STDIO and SSE transports need MCP_FULL_PROXY_ADDRESS

[richardkmichael]

Motivation and Context

#630

The MCP client needs to know the full path to the proxy server when opening the SSE transport for STDIO and SSE endpoints.

The client knows where it expects to see the server, so pass this to the server, to use when building the endpoint.

How Has This Been Tested?

Start nginx as proxy for the backed. Using an nginx proxy: nginx -c inspector.conf (below).

Open the Inspector, set the Inspector Proxy Address to a URL through nginx: http://localhost:9000/mcp/inspector/backend.

# inspector.conf

daemon off;
error_log /dev/stdout info;
events {
}

http {
    upstream inspector-backend {
        server [::1]:6277;
        server 127.0.0.1:6277;
    }

    # Inspector backend only
    server {
        listen 9000;
        listen [::]:9000;

        server_name localhost;

        location /mcp/inspector/backend/ {
            proxy_pass http://inspector-backend/;
            proxy_http_version 1.1;
            proxy_set_header Connection "";
            proxy_buffering off;
            proxy_cache off;
            proxy_read_timeout 24h;
            proxy_set_header X-Accel-Buffering no;
            proxy_set_header Host $host;
            proxy_set_header X-Real-IP $remote_addr;
        }
    }
}

Breaking Changes

No

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

There are other ways to fix this, but my hunch is treating the client as authoritative will be most flexible different deployment scenarios, and also prevent a server-side / client-side configuration mismatch.

Also, I'd add a server test for this, but there are no server tests at the moment. Are server tests desired?

Issue #437 is related, for which there is a fix in a fork, not yet submitted as a PR. It looks like the right approach, appending to the query params from the env. This would close out the current proxy-related issues.