I'm not sure which error you're referring to. When the dialog box the OP posted about comes up, there's an option to play anyway. It's just a warning that single-player data won't sync to the cloud. I've never had to play Skyrim in offline-mode to get around a problem with Steam's servers.
Yes, you can start the game if it can't sync with the cloud, but your latest saves might be missing. I've had both cloud and server connection issues lately.
Disconnect from the internet, restart Steam, and choose offline mod. Easy enough...

I can't go offline if I can't connect to Steam, no matter how many times I restart it.