I can't recall the exact settings to push via ADB, but the Internet Connectivity Check is "easy" to fix. Create a server that's always up that responds with a 301 (or whatever the check expects), and push the address to the phone. Done.

It's a shame that Google's servers are the default, and I wish it were at least called out by Lineage. That said, I doubt they want to cover hosting costs of such a service (although I'd think they'd be fairly minimal).