I think that the path of least resistance for you is going to be to use an Azure DevOps Pipeline. A pipeline is made up of one or more tasks, and there are lots of tasks to choose from.
You can check out source code from a specific branch, build, publish to a slot, swap slots, and copy files using scripted pipelines.
Here is a full list of available tasks. If you would like to copy files from within a repo from one place to another, just take a look at the command line task. The pipeline runs on a Windows or Linux machine (your choice when you choose a Build Agent), so you can use normal DOS copy commands to move files around.