Wrong data path when MCP server mounted as ASGI SSE app under some subpath #464

Describe the bug

When I setup MCP server as a SSE app under some subpath in existing ASGI app, in response to /my/sub/path/sse, I receive wrong endpoint path in data: (missing /my/sub/path prefix). For example if SSE app is mounted under /mcp/test and I call /mcp/test/sse in response I receive:

$ curl -i localhost:8080/mcp/test/sse      
...

event: endpoint
data: /messages/?session_id=e01cc317d1cd48fdb305a8074cee2134

Then MCP client tries to call http://localhost:8080/messages/?session_id=e01cc317d1cd48fdb305a8074cee2134 which does not exist. I believe this is not an issue if MPC server is mounted under root path /, but with subpath it is returning wrong data path.

Side note: There is example of this process in this repo readme but what is the point of mounting app into existing asgi server under /? It will override all routes of existing app, no?

To Reproduce

from fastapi import FastAPI
from mcp.server import FastMCP

# Create a server instance
server = FastMCP("quoting-mcp")

app = FastAPI()
app.mount('/mcp/test', server.sse_app())

call http://localhost:8080:/mcp/test/sse

Expected behavior

$ curl -i localhost:8080/mcp/test/sse      
...

event: endpoint
data: /mcp/test/messages/?session_id=e01cc317d1cd48fdb305a8074cee2134