I nuked the space directory on the server, hoping to trigger a fresh push from the desktop.
Turns out, the desktop also nuked my local space directory. Luckily, I had a backup.
I had been putting .git and .gitignore inside my space directory all this time, before Desktop+ came along, so I never had any issues.
Turns out, the sync server picks up the .gitignore. Instead of doing anything complicated, I simply moved the .git and .gitignore outside the space directory.
I'm posting the solution here in case it helps any newcomers, because the Advanced tab was so well hidden that I only found it by accident.
Hopefully this saves someone else from the same headache.