The same SET USER 640184.571119 command works on my 2012 Brother MFC-9460CDN.
Great write up and reverse engineering of firmware! I hope to see Epson MFC inkjet printers to have similar discovery too (perhaps it literally a prompt away with a link to this article).
This is a really cool little project. Also, the title gave me a strong hit of like mid 2000's internet "hacking" content that I tore through when I was in high school.
I can't tell if any of this gives you more access than the normal http admin interface.