Follow-up from "Add e2e tests for removing current license file and use activation code"

The following discussion from !3738 should be addressed:

• @ddavison started a discussion: (+1 comment)

  We should consider a refactor of this to use a library of some sort. $DOCKER_NAME with no context is nebulous. One might think: Where does the variable come from? What is the name representing? The name of the Docker image for CustomersDot, GitLab?

  This is non-blocking for now, but we should consider extracting this later to some easier-to-use library.