Set container registry auth key via command line/env variable
You mentioned that the auth key is stored in Jenkins credentials and are available as environment variables, if so, then your half task is already done; what you can probably do from here is, use those environment variables to perform docker login (assuming USERNAME and PASSWORD are the Jenkins credentials variable), use the below command to login

docker login your.docker-registry.com -u $USERNAME -p $PASSWORD


This assumes, that your registry has SSL enabled, if so, don't forget to place the registry certificates under the below name on your Jenkin host

/etc/docker/certs.d/your.docker-registry/ca.crt


Post this, you would be able to push image to the registry