@Jake_Walker, unfortunately it's not possible since Buddy requires loads of specific configurations depending on the system. It can be achieved through Buddy CLI only.
To clarify: the containers should be started in a defined order. Some of them should be started once the other ones had already been launched. Otherwise, the application won't work properly.
Unfortunately, docker compose does not give possibilities like that; we'd need to create CLI for this. That's why you won't be able to run it with docker compose.