ci: improve our tests to also trigger a build and push to a registry

[consideRatio]

We have automation setup that test installs the helm chart and ensures pods enter a running state successfully. We should do a bit more.

• Setup a container registry, and grant permissions to k3s that is setup by jupyterhub/action-k3s-helm to pull from it (something like this)
• Make chartpress push images to the container registry as well (something like this)

I think we can be inspired by binderhub's strategies, but that we should aim for a lower complexity:

• z2jh
• binderhub