Not having used DigitalOcean, I cannot say for certain, but based on the tutorial you linked to, I don't think DigitalOcean supports docker-compose in the way you are hoping. (Like you mentioned it does look like you can stand up a VM of your own to serve as the Docker host.)
Compose uses a control plane that sits above the images that you bring up together as a stack. If you're trying to run containers on a remote docker-machine, I suspect you are not allowed to manage that machine's control plane, virtual network interfaces, etc.
I've used AWS Elastic Container Service (ECS) to deploy compose stacks for production. With ECS, you are interacting your own EC2 instance, but Amazon has abstracted away a lot of the management necessary there. AWS has also recently introduced Fargate which adds enhancements for cluster management on top of what ECS offers.
While these are even bigger guns, you might also explore Docker Swarm or Kubernetes for cluster management and orchestration.