Hi,
The Brother DCP-1602 appears to be a USB-only printer. If nothing else works, it could perhaps be worth testing a macOS Catalina 10.15 printer driver even under macOS Big Sur 11. It has been reported that macOS Catalina 10.15 drivers work with some models when Big Sur drivers are not offered.
https://support.brother.com/g/b/downloadlist.aspx?c=us_ot&lang=en&prod=dcp1602_us&os=10060
However, look upon this as a pure experiment. It may not work at all with your printer. There is always a certain risk involved with an experiment. Software conflicts cannot be ruled out. You may have to erase and reinstall everything on the computer. Data may be lost. A hardware problem is unlikely, but it is impossible to guarantee anything. Do not proceed unless you know what you are doing. In any case, make sure that at least three backup copies of important files are available.
Also, if may be a good idea to reset the printing system before any tests.
Reset the Mac printing system to solve a problem - Apple Support
Add a printer on Mac - Apple Support