Thanks. Brother does not appear to offer macOS Big Sur drivers for the MFC-7360N.
However, we have seen before that macOS Catalina 10.15 printer drivers for some models work even under macOS Big Sur.
So, that could perhaps be worth testing with the MFC-7360N as well.
https://support.brother.com/g/b/downloadlist.aspx?c=us_ot&lang=en&prod=mfc7360n_all&os=10060
Look upon this as a pure experiment. It may not work at all. Remember, there is always a certain risk involved with any experiment. Software conflicts cannot be ruled out. A hardware problem is unlikely, but it is impossible to guarantee anything. You may have to erase and reinstall everything on the computer. Data may be lost. Do not proceed unless you know what you are doing. In any case, make sure that you have a minimum of three backup copies of important files before you start.
To begin with, resetting the printing system may be a good idea.
Reset the Mac printing system to solve a problem - Apple Support
If necessary, if you are using USB now (and there is a problem), try a network (Ethernet) connection instead (via one of the LAN ports of a Wi-Fi router).
Add a printer on Mac - Apple Support