Berksfile source doesn't recognize self-hosted Chef server
Solution 1:
Turns out that the source in a Berksfile is actually the URL of a Berkshelf API Server, not a Chef Server API.
After doing gem install berkshelf-api and configuring the config.json to use my Chef Server as an endpoint, I was able to run berks-api and then use port 26200 on my Chef Server as the target of the source in the Berksfile.
From here, berks install and berks upload both worked.