Brother MFC-8840DN and OS X Catalina
After upgrading, the existing printer instances (CUPS and BR-Script3) worked as they had in OS X Mavericks (and earlier).  Both suffered from the inability to remove the job from the queue after printing, but this isn't related to Catalina.

I created a new printer instance, using the Brother MFC-8840D BR-Script3 driver and everything works as expected, including clearing the queue.

Note: OS X does not recognize the device's scanner feature, so I cannot use Preview to Import from scanner....