There are several solutions available, depending on your private cloud context one is more useful then the other.
Packaging: Using Mendix Studio Pro you should be able to create a package. Alternatively Mendix Deploy API can be used to create a package.
Deploying: Mendix Deploy API could help you with this.
Probably Mendix Deploy API can be scripted in your favorite private cloud hosting portal to ease repeating scenarios.
There are 2 main ways to do it outside of the Mendix platform.
These will produce a deployment package with MDA extension. In private cloud you probably better use https://github.com/mendix/docker-mendix-buildpack to build a docker image and then deploy it to any container orchestration system like ECS, Kubernetes, docker, etc… This buildpack accepts MDA and project/model source as input.
Thanks for your feedbacks, and I will test the solutions on my side.