You can actually take a copy of your character and play it offline. There's a locally stored copy of your character somewhere in the Diablo II directory. Search for your character's name. Copy the .d2s file to your save folder and voila. Battle.net isn't going to have any of your single player progress on your character though. Essentially you'll be playing 2 characters who started out the same, have the same name, but are completely separate.
There's also the option of playing 'Open' Battle.net, if that's not yet abolished. You can use any character you like there.