Printer cartridge is LC422. According to the list on the "tintenprofi.ch" website, the cartridges are compatible with your device. Unfortunately, I can't help you with your problems with "Update" and "Climate glue".

The cartridges from the old MFC-J6910 are NOT compatible with the new MFC-J6940.
Both devices are also supported under Windows 11. If there are problems after the Windows upgrade, simply uninstall the complete driver, restart the computer and download and install the Windows 11 driver 8complete software package) from the Brother website.

I have a Brother MFC-J6940DW which works perfectly under Windows 11 Pro, the cartridges are LC422-LC422XL.

I have no problems with Apple computers: Download the latest driver for Windows from brother.ch, then it should also work with Win 11.