I do something like this. I run wireguard in a container along with dante-server (a socks proxy daemon). I then configured a Firefox profile to connect to the socks daemon running in the container.
This way I have a single browser profile that is routed through Mullvad while everything else works normally.
I'm very upset by this decision. I've been using Signal as my SMS app for a very long time.
Messages that I would have sent via SMS currently will automatically get sent via Signal if the person I'm sending to has started using Signal without my knowledge. This has happened in several instances where I was pleasantly surprised to see a friend had started using Signal. Now that I'm forced into a separate SMS app, this will no longer be a possibility. I certainly won't be firing up Signal to see if a contact has joined before sending them an SMS.
I finally set up two factor auth in gmail using FreeOTP and have Fairemail (fdroid version) configured using an app password. (I run LineageOS with no Google services installed or account setup so the OAuth method isn't an option for me)
This way I have a single browser profile that is routed through Mullvad while everything else works normally.