I'm having the same issue, so I will try to fix it later and get back to this post if any of these methods work.
EDIT:
I tried both methods. The Arch one did nothing, while the xdg-settings one did set Mullvad Browser as my default, but when I click a link the browser will open on the homepage anyways. So yeah, seems like we gotta wait for this to get fixed by the Mullvad devs.