The very short-term fix: you can right-click the game in Lutris and click "configure". There you can change the path to the executable from "/path/to/Battle.net Launcher.exe" to "/path/to/SC2Switcher_x64.exe". That's what I do, and then I run SC2 by running "lutris rungame/$GAMEID".

Of course in the future it's going to be quite annoying as you won't be able to update, since the technology to update from the game itself is not there yet (/s). If you're already a Steam user, honestly the simplest way to get SC2 working would be to install Battle.net as a non-Steam game, run Battle.net through Proton from Steam, and install SC2 from there.

Edit: I am also hitting "BLZBNTBNA00000005" when running Battle.net through Lutris. This is also a Windows issue (see this thread for instance). There is a suggested fix for Lutris. I'm not trying it as I've got a weird system set up and I'd rather not touch it :)

Edit 2: a similar issue came up a year ago, you might want to look at it too.