FastMCP: streamable_http_app() silently breaks when BaseHTTPMiddleware is added

Description

Adding a Starlette BaseHTTPMiddleware (e.g., for auth) to FastMCP.streamable_http_app() silently breaks the MCP server. Every request crashes with ClosedResourceError — the client sees "peer closed connection without sending complete message body."

Reproduction

from mcp.server.fastmcp import FastMCP
from starlette.middleware.base import BaseHTTPMiddleware

mcp = FastMCP("test")

# Any BaseHTTPMiddleware — even a no-op passthrough
class AuthMiddleware(BaseHTTPMiddleware):
    async def dispatch(self, request, call_next):
        return await call_next(request)

mcp.streamable_http_app().add_middleware(BaseHTTPMiddleware, dispatch=AuthMiddleware)

Run the server, send any MCP initialize request → ClosedResourceError.

Root cause

BaseHTTPMiddleware wraps the ASGI receive/send channels in a way incompatible with SSE streaming. This is a known Starlette limitation (encode/starlette#919), but FastMCP users hit it naturally when they reach for the obvious auth pattern.

Expected behavior

At minimum: warn at startup when a BaseHTTPMiddleware subclass is detected on a streamable HTTP app. Ideally: document prominently that users should use FastMCP's own Middleware class (fastmcp.server.middleware.Middleware) instead of Starlette's BaseHTTPMiddleware.

Workaround (raw ASGI middleware)

from starlette.types import ASGIApp, Receive, Scope, Send

class RawAuthMiddleware:
    def __init__(self, app: ASGIApp):
        self.app = app

    async def __call__(self, scope: Scope, receive: Receive, send: Send):
        # auth check here — read headers from scope
        await self.app(scope, receive, send)

mcp.streamable_http_app().add_middleware(RawAuthMiddleware)

Environment

• mcp / FastMCP from modelcontextprotocol/python-sdk
• Starlette (any version — this is architectural, not a regression)
• Python 3.12

Comienza en el punto de entrada FastMCP.streamable_http_app() e inspecciona cómo se registra el middleware alrededor de la aplicación HTTP transmitible. Reproduce el fallo con el ejemplo no-op de BaseHTTPMiddleware y determina después cómo deberían funcionar la detección durante el inicio y el comportamiento de las advertencias. Se considera terminado cuando el fallo se muestra claramente y se dirige a los usuarios hacia Middleware de FastMCP o la solución alternativa de ASGI sin procesar.