Thank you. Additional information from Brother does not appear to help much.
https://help.brother-usa.com/app/answers/detail/a_id/176091
AirPrint is dependent on Bonjour. Any change if you try the following power-cycling sequence?
Power off the Wi-Fi router, the printer(s), the computer, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer(s). Wait a minute or two. Power on the computer. Finally, power on all other devices/printers.
If you have access to an iPhone/iPad, is it possible to print from there?
Use AirPrint to print from your iPhone, iPad, or iPod touch - Apple Support
If absolutely nothing else works, you could possibly try the macOS 10.15 Catalina printer driver 1.6.0 (the same link as in my previous message). This is not AirPrint. Probably a good idea to again reset the printing system. Could perhaps be tested with USB as well. It would be entirely experimental. It may not work at all with your printer and macOS Monterey. Before doing anything, make sure that you have backup copies of all important files (documents, photos, et cetera).