fix(mcp): mount session manager via Starlette Mount to avoid double response.start

StreamableHTTPSessionManager is an ASGI app — it sends its own
http.response.start and http.response.body messages via the scope/receive/send
triple. Wrapping it inside a @app.api_route FastAPI handler causes FastAPI
to try to finalize the response after the handler returns, emitting a
second http.response.start. uvicorn rejects this with:

    RuntimeError: Unexpected ASGI message 'http.response.start' sent,
    after response already completed.

Every POST /mcp raises, making the server unreachable from any MCP client
(tested with Claude Code 2.x against mcp==1.27.0).

Fix: mount the session manager as a Starlette Mount so it owns the
response cycle directly. Auth middleware still applies because
add_middleware attaches at the app level, wrapping all mounted sub-apps.

mcp/app/main.py

@@ -1,7 +1,8 @@
 from contextlib import asynccontextmanager
-from fastapi import FastAPI, Request
+from fastapi import FastAPI
 from mcp.server import Server
 from mcp.server.streamable_http_manager import StreamableHTTPSessionManager
+from starlette.routing import Mount
 
 from .auth import ApiKeyMiddleware
 from .backend_client import backend
@@ -28,15 +29,20 @@ async def lifespan(app: FastAPI):
     await backend.stop()
 
 
-app = FastAPI(title="Homelable MCP", lifespan=lifespan)
+# Mount the session manager as an ASGI sub-app instead of wrapping it in a
+# FastAPI @app.api_route handler. Wrapping it in a route handler causes
+# FastAPI to send http.response.start after the session manager has already
+# started the response, raising `RuntimeError: Unexpected ASGI message
+# 'http.response.start' sent, after response already completed` on every
+# POST /mcp — which makes the server unreachable from any MCP client.
+app = FastAPI(
+    title="Homelable MCP",
+    lifespan=lifespan,
+    routes=[Mount("/mcp", app=session_manager.handle_request)],
+)
 app.add_middleware(ApiKeyMiddleware)
 
 
-@app.api_route("/mcp", methods=["GET", "POST", "DELETE"])
-async def mcp_endpoint(request: Request):
-    await session_manager.handle_request(request.scope, request.receive, request._send)
-
-
 @app.get("/health")
 async def health():
     return {"status": "ok"}