Connecting screen won't go away in build mode

@SageyTea  Try F1, which would normally put you into live mode.  Other than that, I'm not sure there's anything you could do.  But you could always take your computer offline.  Clicking cmd-tab will allow you to switch to another app, which should give you access to wifi settings.

Other than that, if you don't mind potentially crashing the game, which would be the only other option anyway, you could bring up the Activity Monitor, in Applications > Utilities, and try to find the process that's attempting to connect.  I don't know what it would be called but would look for something that starts with EA or Origin.  If you force-close that (double-click to see the option), perhaps the game will stop trying to connect.  But this is pure speculation.

Finally, there might be a way to recover your progress if you do have to force-quit the game, which you can do with cmd-optn-esc.  When Sims 3 crashes or closes not in the normal way, it leaves behind files in Documents > Electronic Arts > The Sims 3 > CurrentGame.sims3.  These files might translate into a save you can load.  I would absolutely not play this save forward, but you may be able to load it long enough to save your in-progress build to the bin.  If you'd like to try, drop the CurrentGame.sims3 folder into Saves and see what happens.