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Reading more of the documentation, this article helped me: http://dokku.viewdocs.io/dokku/deployment/user-management/#adding-deploy-users
I’m not sure why I was able to ssh in, but unable to git push with the same ssh key, but either way passing a fresh copy of my ssh key to the server did the trick:
# from your local machine
# replace dokku.me with your domain name or the host's IP
# replace root with your server's root user
# USER is the username you use to refer to this particular key
cat ~/.ssh/id_rsa.pub | ssh root@dokku.me "sudo sshcommand acl-add dokku USER"