An ASGI deployment doesn’t help you here indeed. It’s a definition for being a websocket server, not a client.
I’d write a management command that uses asyncio.run in its handle() to connect to the websocket. The websockets library may be of use.
ORM operations to save data to your DB should be done in a sync function wrapped with sync_to_async, and you’ll need to take care that you close database connections appropriately.