SSE reconnect does not include Last-Event-ID header

[Yeaury]

Describe the bug
When using mcp-inspector with a MCP server over Streamable HTTP (SSE), the client does not send the Last-Event-ID header during reconnects. As a result, the server cannot resume the stream from the last delivered event, which breaks the MCP protocol’s resume functionality.

To Reproduce
Steps to reproduce the behavior:

1. Start a MCP server that supports SSE and emits events with an id: field.
2. Connect to the server using mcp-inspector.
3. Interrupt the connection (e.g. restart the server or disable/re-enable network).
4. Inspector automatically reconnects, but the HTTP request does not include the Last-Event-ID header.

Expected behavior
On SSE reconnect, mcp-inspector should include the Last-Event-ID header with the last received event id, so the server can continue sending events from the correct position.

Logs
Observed server logs show reconnect attempts, but Last-Event-ID is missing in the request headers.

Additional context

• MCP specification for Streamable HTTP/SSE explicitly describes support for Last-Event-ID.
• Without this header, SSE resume functionality cannot work and events may be lost.
• Related discussion: [#679](Negotiated mcp-protocol-version gets lost at the proxy layer #679)