Something to try, though I don’t have anything close to that configuration to test with:
That printer looks to be a dozen years old, and well predates AirPrint, which means it’ll need vendor drivers, or maybe the IPP/IPPS path will work.
To try using CUPS and IPP/IPPS printing, you will need the IP address information for the printer, and the following steps to enable and access CUPS:
- Launch Terminal app
- Paste the following command into the command line, and press return:
cupsctl WebInterface=yes
- Quit Terminal, and Launch Safari
- Visit localhost:631 and try to add your Brother printer there, using IPP/IPPS printing.
AirPrint is related to IPP/IPPS, and (like IPP/IPPS) is intended to eliminate the need for vendor printer drivers.
This is also why AirPrint is usually a feature usually recommended when selecting a printer for Apple platforms.
CUPS quick intro, and adding a printer using the web interface: https://www.cups.org/doc/overview.html
Detailed technical CUPS network configuration info: https://www.cups.org/doc/network.html
Brother printers are usually pretty good even without drivers, and are particularly good when AirPrint is available.