Is it possible for an extension to be present without it triggering Firefox’s “installation” flow?

I agree that straight up using Tailscale would likely be easier. But to answer your question, you’re looking to “push routes” because what you actually want to do is “route” but that’s kinda hard to Google haha. This looks maybe promising: https://forums.freebsd.org/threads/wireguard-how-to-route-another-subnet-through-it.89744/