Well, the 2360dw driver was recognized and I thought I was good to go, but it got stuck "processing" when I tired to attempt to print. So I removed the printer again and went back to the starting spot. This time it was not recognized.
I get it about rebooting. I know that and have done that repeatedly throughout this process.
I guess I've been lucky that I've never had any trouble with this printer until now.
With all the digging and trying and failing I've done in this situation, I think this meets my criteria for a bug. So I filed a bug report.
I hope some easily repeatable Brother solution comes along. I love this distro and want to make it my home.