Apparently Mullvad VPN adds some firewall rules that break traffic forwarding.
Perhaps there’s a way to prevent the VPN messing with the firewall, but they don’t offer any free plan, so it’s problematic to test and investigate.
As a workaround, you can try importing the VPN profile to NetworkManager and avoid using their client app.
On the other hand, a shared host bridge can be configured like this:
How to make a Virtual Machine present itself to my home network as if it were another physical computer - #7 by vgaetera