The SSE server in fastmcp currently operates as a Starlette app. In addition to running it as a standalone server, it would be beneficial to provide an option to mount it to an existing ASGI server.
Suggested solution
Currently, the run(transport="sse") method and the run_sse_async() method include the code for actually serving the application. It would be better to separate this functionality and allow users to obtain an instance of the SSE server app via a method like mcp.sse_app() or mcp.starlette_app().