Just continue through the message and it works. I had the same issue on switch. If you just press okay when you get the Airplane mode message it will continue to the offline character screen.
If this doesnt work hotspot your phone until you get to where it says Connecting to Bnet servers then turn it on Airplane mode. It will immediately go to offline characters. It makes you think you have to turn it off, but you can continue if you just press Okay. Hope this helps!
I had a similar problem with the PS4 version. I had to press the home button then go back to the game to see the access code to type it in to the battle.net app on my PC. Hopefully that helps.
This is why blizzard needs to remove the fraud 30 day DRM, nintendo gave me my money back because this games “ Single player “ quit functioning after 30 days, this game is being sold under a perpetual license not a subscription license, games as a service is fraud.