Yay! A Windows installer wizard starts up. Looks like "Run anyway" was the right choice. I complete the installer's steps, and click "Finish" (leaving "Run Whatsapp Portable" unselected, as something I saw earlier in PlayOnLinux suggested avoiding launching app on finish).
Entire screen goes black for a bit. Eventually screens returns to as before, including the installer again waiting for me to click Finish.
I click it again, and the installer goes away, normal screen this time.