Since you wish to play a game offline anyway, what you can do is once game is loaded and ready to play in offline mode, open up Task Manager and completely shut down the EA app and EA Background Services program - Shutting down latter first will also shut down EA app. Problem removed. No more EA app and Services trying to phone home when supposed to be offline. Play the game without any more errors from app popping up and interfering.
Also when finished playing game and shut it down, the next time you need to load up the EA app it will not be asking for your login details. So that's annoyance number two also removed. It's all good.
Another method might be to open Settings on the EA app and shut down the Background Services so it does not auto run. It is switched on by default. Perhaps the EA app itself can then run unhindered by connection errors in offline mode. Although I haven't tried this second method as the first one above does the trick for me... so far. Task Manager is your friend.
Edit - Just now tried the second method and I don't trust it to hold water. Loaded app, went to Settings, turned off Background Services and Notifications - offline who needs 'em? - shut down app and opened Task Manager to manually shut it all down properly, thoroughly - as it will continue to run in background anyway by default even if you believe you've closed the app. They all do this, Steam, Epic, EA certainly not alone in continuing to run even if you think you've shut it down. Then loaded app fresh again. Checked Settings, Background Services remain closed, but opening Task Manager to spy they are actually running as normal. Who's pulling whose leg? Ergo, no trust in this second method holding out. "Nuke it from orbit" with Task Manager as in method one. "The only way to be sure."