### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

When LiteLLM's proxy connects to an upstream MCP server via Streamable HTTP, it receives a `mcp-session-id` header in the `initialize` response but does **not** include that header in subsequent requests (`tools/list`, `tools/call`). Each follow-up request arrives at the upstream server without a session ID - the server treats it as a brand-new uninitialised session, and tools never load.

Expected: after receiving `mcp-session-id: <uuid>` in the `initialize` response, LiteLLM should echo that header back on all subsequent requests to the same upstream server, as required by the [MCP Streamable HTTP spec](https://spec.modelcontextprotocol.io/specification/2024-11-05/basic/transports/#streamable-http).

This is the **outbound MCP HTTP client** behaviour (LiteLLM connecting to upstream servers), distinct from #20242 which concerned LiteLLM's own server-side session manager.

### Steps to Reproduce

1. Start any stateful Streamable HTTP MCP server - e.g. `isokoliuk/mcp-searxng` with `MCP_HTTP_PORT` set, or Supergateway in `--stateful` mode. 2. Add…