You don't need a "network kill switch" with wiregaurd, you might be using the openVPN option which mullvad also provide for compatibility. Because wiregaurd is stateless you don't have to worry about stuff leaking through while physical layers go up and down, you can just leave the wg interface up and keep hoping around safely... I literally haven't taken my current wg connection down in days, yet my computer is put to sleep every night.
If you use Linux you don't even need an app (not Firefox's or Mullvad's), you can just pop one of the wiregaurd configs (mullvad.net can generate them for you) into /etc/wiregaurd and then use the super simple wg-quick cli interface to bring it up. You can also tell systemd to bring up a specific interface at startup with one line.