It'll be wonderful was possible selecting a specific network path to Firefox browses on the Internet. I've more than one Internet connection in my home, but sometimes I use a VPN connection established in my router via another IP address on my computer. That way, It'll be interesting was possible to set that connection to private mode too.
Privacy is all on Firefox.
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i'd say you setup a local proxy that is able to switch between the 2 network interfaces and configure firefox to use that proxy .. maybe also install something like Foxy Proxy Basic/Standard to easily switch.