After digging through the pip issue 3939 linked by @muon in the comments above and then the PEP-508 specification, I found success getting my private repo dependency to install via setup.py using this specification pattern in install_requires (no more dependency_links):
install_requires = [
'some-pkg @ git+ssh://git@github.com/someorgname/pkg-repo-name@v1.1#egg=some-pkg',
]
The @v1.1 indicates the release tag created on github and could be replaced with a branch, commit, or different type of tag.