[BUG] router prefix is ignored

[danieleds]

Describe the bug
Sometimes, the FastAPI router itself doesn't know where it's going to be mounted. This makes it impossible to mount a FastApiMCP instance under a given path.

Example

from examples.shared.apps.items import app # The FastAPI app
from examples.shared.setup import setup_logging

from fastapi import APIRouter
from fastapi_mcp import FastApiMCP

setup_logging()

other_router = APIRouter()  # No prefix specified here
app.include_router(other_router, prefix="/other/route")  # Prefix is specified in the include_router call

mcp = FastApiMCP(app)

mcp.mount(other_router) # This will expose the endpoint in the `other_router` at the root, not under `/other/route` as expected 
mcp.mount(other_router, mount_path="/other/route") # This is not the expected usage of the mount_path parameter, and does not work
mcp.mount(other_router, mount_path="/other/route/mcp") # This is not the expected usage of the mount_path parameter, and does not work

if __name__ == "__main__":
    import uvicorn
    
    uvicorn.run(app, host="0.0.0.0", port=8000)

Basically, unless the FastAPI router is instantiated with a prefix, it is impossible to mount a FastApiMCP in a subpath.