An error occurred starting StarCraft II. Please try again. (18.04)

In January 2019 I switched to Ubuntu 18.04. To my great relief I was able to easily install StarCraft II through Wine. That was the end of the story until now, when I got a new computer, but this time I’m having a lot more trouble. Thanks to this very site, I was able to install Battle.net (though I was having quite a bit of trouble with it for other reasons), but then, after installing StarCraft II, I couldn’t get it to start. When I try to start StarCraft II, I get an error message saying ERROR: Game Initialization Failed: I followed by another error message saying StarCraft II: An error occurred starting StarCraft II. Please try again. (Error 2:1), and then nothing.

If I tell Battle.net to try running the 32-bit version, the error messages don’t appear, but it still doesn’t actually start (and the latter message will appear if I kill SC2.exe). I can’t find much useful information online about either of these messages, so I’m asking about it here instead.