The doctl integration is just a convenience. You can login to the registry without it. To do so, use a DigitalOcean API token for both the username and password:

docker login -u $DO_TOKEN -p $DO_TOKEN registry.digitalocean.com

Additionally, you can retrieve a .docker/config.json file that you can use with Docker using the API directly:

curl -X GET -H "Content-Type: application/json" \
  -H "Authorization: Bearer $DO_TOKEN" \
  "https://api.digitalocean.com/v2/registry/docker-credentials 

See: https://developers.digitalocean.com/documentation/v2/#get-docker-credentials-for-container-registry