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

Start at the FastMCP.streamable_http_app() entry point and inspect how middleware is registered around the streamable HTTP app. Reproduce the failure with the no-op BaseHTTPMiddleware example, then determine how startup detection and warning behavior should work. Done means the failure is surfaced clearly and users are directed toward FastMCP's Middleware or the raw ASGI workaround.