A couple of days after posting my question, I had the remaining computers leave the homegroup, and then created a new homegroup password, and we re-joined the "new" homegroup.
The ghost user didn't immediately go away, but lost its personalized icon.
The ghost user is now gone from the homegroup. I hope it doesn't show back up again, but for now, it looks like the issue is resolved.
Thanks to everyone who read the post.